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last marked /below. 

_ L __ 




by the late 

Formerly Wykeham Professor of Logic 
In the University of Oxford 

Reconstructed from Notes 
taken by his Pupils 

Edited by Errol E. Harris 
Foreword by Sir David Ross 



This book is copyright under the Berne Convention. 
Apart from any fair dealing for the purposes of 
private study, research, criticism or review, as per- 
mitted under the Copyright Act 1911, no portion may 
be reproduced by any process without written per- 
mission. Enquiry should be made to the publisher. 


in n 1 12 pt. Pilgrim type 



During the early 1930*5, Harold H. Joachim, as Wykeham 
Professor of Logic at Oxford, delivered the Logical Studies 
as a course of lectures extending over two terms of the 
academic year, and in the third term he delivered a set 
of lectures on the Regulae of Descartes. The Logical Studies, 
have long since been published, but the manuscript of the 
Descartes lectures was lost, and there is reason to believe 
that it was accidentally destroyed with certain domestic 
papers of no philosophical importance. With the extinction 
of all hope of finding the original manuscript it seems 
fitting that, rather than submit to the complete loss of this 
work to the world of scholarship, an attempt should be 
made to reconstruct it from the notes taken by some of 
those who heard the lectures. Two sets of notes (all, so 
far as I can ascertain, that are available) have been used 
for this purpose: those of Mr. John Austin, now White's 
Professor of Moral Philosophy at Oxford, and my own. 
Let me at once record my gratitude to Professor Austin 
for the use he has allowed me of his excellent set of notes. 
It was my own endeavour as a student to get down, so far 
as was physically possible, every word of Joachim's 
lectures, but Professor Austin, by adopting a more tele- 
graphic style, succeeded in recording even more of the 
substance and detail of the lectures than I had. 

In attempting to reconstruct what Joachim had actually 
written, I have tried not to omit anything of the least 

importance which is contained in either set of notes. 
Where they were verbally identical little difficulty pre- 
sented itself and it seemed safe to presume that here one 
had very nearly, if not exactly, what the lecturer had 
said. Where the notes differed verbally but not in sense, 
I have adopted, whichever rendering seemed to me to ex- 
press better the thought of the author as I remember it. 
If in this respect there was little to choose between the 
two versions, I have adopted the one which (with least 
adjustment or modification) would read better. But both 
versions are no more than students' notes, taken in lectures 
and written under pressure of time, and it has been neces- 
sary throughout to make minor corrections and to supply 
omissions, both in order to produce a continuous prose 
style and in order to clarify the meaning. That these cor- 
rections and intercalations are even near to what Joachim 
wrote or would have written (had he lived to revise the 
work) there is, of course, no means of knowing, but I 
have constantly kept in mind the sort of thing that I 
remember he used to say as well as what he has written 

If Joachim had lived it is almost certain that he would 
not have permitted the publication of his own notes before 
he had carefully revised them more than once. He would 
probably have corrected and modified them or even re- 
written them either wholly or in part. He would pre- 
sumably have written a conclusion to avoid the abrupt- 
ness with which these lectures end. In short, the publica- 
tion of the present version is to be tolerated only as a 
lesser evil than the total loss of the thought and work on 
this subject of one of the most erudite and careful 
scholars of the last generation. 

It is, further, regrettable that the appearance of this 
work of Joachim's should have been so long delayed. But 
it was necessary to make sure that no original, nor any 
better and more authentic version, was ever likely to be- 

come available before resorting to the very inferior sub- 
stitute of students' notes. In the meantime further research 
has been done on the subject of Descartes's method and, 
in particular on the Regulce. New editions (e.g. by Leroy 
and Gouhier) have appeared and a number of works in 
French, besides Dr. L. J. Beck's admirable book in English. 
The reader will have to bear in mind that Joachim's 
lectures predate all these, and that much which he wrote 
in 1930 (or thereabouts) he would undoubtedly have re- 
considered had he lived to read and know of this more 
recent work. But the extent to which he would have modi- 
fied his own writing cannot be known. I have, therefore, 
made no attempt to edit the lectures in the light of later 
research. It is not impossible that scholars writing after 
Joachim would themselves have been influenced by his 
thought had it been published in time. It seems to me that, 
in the circumstances, it is important to make this little 
work available in the best form possible without more 
ado, as something of lasting value to philosophers in 
general and to Cartesian scholars in particular. It is not 
for me (or, as I see it, for even more competent persons) 
to correct what Joachim wrote in the light of later 
scholarship as it cannot be known whether, to what ex- 
tent, or in what way, he would have done so himself. 

There is one further consideration. The contemporary 
idiom in Anglo-Saxon philosophy is so utterly different 
from that in which Joachim wrote and thought only 
twenty years ago, that many students may question the 
value of this publication. That the work is at least of 
historical interest, both in itself and as a contribution to 
the history of philosophy no genuine scholar will deny. 
But it is of even greater value than that in as much as it 
is a contribution (far greater than its physical size sug- 
gests) to a kind of philosophy which, if it is at present 
not so widely practised, nevertheless, has in it much sound 
substance and significance, and one which may well re- 

turn to fashion in the not far distant future. There are 
signs that, for all their vigour, contemporary empiricisms 
have reached the limit of profitable development. Even 
professional philosophers may shortly be forced to look 
elsewhere for fruitful means of advance, if only by the 
pressure exerted upon them by the progress of the natural 
sciences, the direction of which seems to point to a 
philosophy very different from that now current, or by 
the demand for a re-interpretation of human experience, 
which the inescapable course of international politics 
makes upon them. 

It remains only for me gratefully to acknowledge the 
assistance of the University of the Witwatersrand whose 
subvention has made the publication of this book possible. 


The University of the Witwatersrand 



page 13 

page 19 


page 62 

page 100 



I have gladly accepted Professor Harris's invitation to me 
to write a brief introduction to his edition of Professor 
Joachim's lectures on Descartes's Rules for the Direction 
of the Mind. I had the good fortune of knowing Professor 
Joachim for the last forty years of his life. His philosophi- 
cal views have been admirably described and discussed in 
Joseph's memoir of him in vol. 24 of the Proceedings of 
the British Academy, and any who wish to see a sym- 
pathetic and yet critical account of them cannot do better 
than read that memoir. My own attitude towards 
Joachim's philosophy is not unlike that of Mr. Joseph. I 
cannot accept Joachim's coherent theory of truth. But I 
greatly admire its scholarship and exactness of his ex- 
position, whether the philosopher he was concentrating on 
was Aristotle or Descartes or Spinoza; in each of those 
fields of study, he was a master. In the interpretation of 
Aristotle the student's first task is to discover what 
Aristotle actually wrote, and that involves the careful 
study of the manuscript tradition and of the ancient com- 
mentators, and the establishment of a correct text, in 
which even the punctuation is a matter of importance. 
Joachim's scholarship and skill in all this I was able to 
observe at the weekly meetings of the Oxford Aristotilian 
Society from 1900-1914, and in reading his recently 
published commentary on the Nicomachean Ethics. 

His work on Descartes and on Spinoza exhibits the same 
qualities of scholarship and of philosophical acumen. His 
book on Spinoza is the best study in the English language 

at any rate of this great philosopher. Professor Harris has 
been able to reconstruct, from his own notes and those 
of other pupils, Joachim's lectures on Descartes, the 
original manuscript of which was unfortunately lost. In 
this reconstruction some of the nuances of his exposition 
will no doubt have been lost, but enough remains to make 
Professor Harris's edition a most valuable contribution to 
the study of Descartes. 

Mr. T. S. Eliot, who was Joachim's pupil at Merton 
College, has borne testimony to the close but luminous 
style (to quote the words of the present Warden of 
Merton) of his writing; and any of his philosophical 
colleagues at Oxford who are still alive would bear testi- 
mony to the clarity and firmness (combined with exquisite 
courtsey) with which he would expound his views or 
criticise those of others. Professor Harris's book will revive 
in our minds the impression which Joachim made in our 
oral discussions. 

W. D. Ross. 


Charles Adam suggests that the Regulae ad Direction- 
em Ingenii, the unfinished dialogue, La Recherche de la 
Verite par la Lumiere Naturelle, and Le Monde are re- 
lated to Discours de la Methode, Meditationes and 
Principia Philosophiae as the first crude sketch for a 
finished masterpiece. 1 This view is fully confirmed by 
comparison of the Regulae with the Discours. 

The Discours was first published in 1637 (when 
Descartes was 41 years old) as the exposition of the 
general principles of the Cartesian method. It is 
masterly in conception and exposition, as well as in 
its lucidity and coherence. The Regulae, in contrast, 
was written probably in the winter of 1628-9, or even 
earlier, and is unfinished 2 and in many ways imperfect. 
The work is immature and was probably left un- 
finished because of its defects. Descartes is still, in some 
respects, feeling his way. His exposition is often con- 
fused and rambling and is sometimes inconsistent. He 
is more dogmatic than in the later works, in which 
some of the doctrines here stated are rejected or 
modified. The work is also immature in form. For 

1 Cp. Adam & Tannery, X, pp. 530-2 and XII, pp. i46ff. 

2 There were to have been 36 rules (cf. Reg. xii), but only 23 exist, 
and the last three lack explanatory expositions. 


example, Rule xi 1 merely repeats Rule vi 2 in a com- 
pressed form; the long autobiographical passage at the 
end of the exposition of Rule iv 3 seems to have been 
added as an afterthought and is ill-fitting in that place, 
and Rule viii combines, without reconciliation, a rough 
draft with a more finished but incompatible version. 

Owing to these defects, most editors pass over the 
Regulae with brief notice, and the accepted and auth- 
oritative account of Descartes's method is based on the 
Discours. But a detailed study of the Regulae is in- 
structive as well as interesting, if for no other reason 
than that it constitutes the first material for the exam- 
ination of the Cartesian conception of vera mathesis. 
Though nothing emerges in the Regulae which is ir- 
reconcilable with the traditional exposition of the 
method, yet it presents difficulties which do not ap- 
pear in the Discours, and so provides a fuller under- 
standing of Descartes's teaching. 


Descartes died at Stockholm on the nth February, 
1650. Two inventories of his papers were made, one 
at Stockholm, on the i4th of February, of papers he 
had brought into Sweden, and the second at Leyden, on 
the 4th of March, of papers left in Holland. M. Jean de 
Raey, a Professor at Leyden and friend of Descartes, 
testifies that these were few and of small importance 

1 A. and T., X, pp. 409-410. 

2 Ibid., pp. 384-7. 

3 Ibid., pp. 374-9- 

4 Cp. A. & T., X, pp. 1-14, 351-7, 486-8, 


as Descartes took the best with him to Sweden. 1 Of 
the earlier inventory, made 3 days after Descartes's 
death, two manuscript copies survive. 2 There are 23 
rubrics and the Regulae is mentioned under Rubric 
F : 'Neuf cahiers reliez ensemble, contenans partie d'un 
traite des regies utiies et claires pour la direction de 
1' Esprit en la recherche de la Verite'.* 

All (except domestic papers) that were contained in 
this inventory were entrusted to M. Chanut, French 
ambassador to Queen Christina of Sweden; he convey- 
ed them to Paris, and, being too busy himself to pub- 
lish any of the manuscripts, entrusted all or most of 
them to his brother-in-law Clerselier 4 who was also a 
friend of Descartes. 

Clerselier published three volumes of letters in 1657, 
1659 and 1667. He also published, in one volume, a 
treatise on Man (L'homme), in 1664, and in the 
second edition (1677) added 'Le Monde (ou Traite de la 
Lumiere). In a preface to Vol. Ill of the letters, Cler- 
selier says that there are still more than enough MSS. 
to make another volume of fragments and offers them 
to anyone who is willing to edit them. Nobody accep- 
ted the offer and Clerselier died in 1684 leaving them 
unpublished. He passed the MSS. on to the Abbe Jean 
Baptiste Legrand who set out admirably to produce a 
complete edition of all Descartes's posthumous works, 
but he died in 1704 before the edition was finished, 
leaving its completion to Marmion. But he too died, 

1 A. & r, v, p. 410. 

2 A. & T., X, pp. s-12. 

3 Ibid. p. 9. 

4 Vide ibid., p. 13. 

in 1705. There does exist in Paris a richly annotated 
copy of Clerselier's three volumes with marginal notes, 
additions and corrections in the hand of Legrand, 
of Marmion and of Baillet (Descartes's biographer). 

After Marmion's death the papers reverted to Leg- 
rand's mother and it is not known what became of 
them thereafter. The MS. of the Regulae has thus van- 
ished, but a number of people saw and used it : (i) The 
second edition of the Port Royal Logic (Arnaud and 
Nicole, I664 1 ) contains a long extract from Rules xiii 
and xiv translated into French. This the editors owed 
to Clerselier who lent them the MS. 2 (ii) Nicolas 
Poisson in his Remarques sur la Methode de M. Des- 
cartes (1670) says that he saw the MS. 3 (iii) Perhaps 
Clerselier also showed it to Malebranche, who himself 
published a work in 1674-5 under the same title as 
Le Recherche de la Verite. (iv) Baillet, in his Life of 
Descartes (1691) quotes the Regulae freely and uses 
them in many places. He says Legrand lent him the 
MS. 4 (v) A note discovered at Hanover in Leibniz's 
handwriting says that with Tschirnhaus he visited 
Clerselier, who showed them both certain MSS of Des- 
cartes including Le Recherche de la Verite and '22 rules 
explained and illustrated'. 5 

At least two MS. copies of the Regulae were made 
in Holland. One was bought by Leibniz in September 
1670 from Dr. Schuller for the Royal Library at Han- 
over, where it now remains. It is an inferior version 

1 The first edn. appeared in 1662. 

2 Vide A. & T., X, p. 352. 

3 Ibid. 

4 Cp. ibid. 

5 Cp. ibid. pp. 208-9. 

with many omissions and mistakes. Leibniz (not at the 
time knowing Descartes's handwriting) says that it is 
in the author's own hand, but this is not the case. The 
other copy is much better and probably belonged to 
Jean de Raey. It was used for the Flemish translation 
of 1684 by Glazemaker and served, no doubt, as the 
original for the Amsterdam edition of Opuscula 
Tostuma of 1701. The version given by Adam and 
Tannery is based primarily on this, which they refer 
to as 'A', correcting it at times from the Hanover MS. 
(for which the reference is H). 


Adam suggests the winter of 1628-9 as the time 
when Descartes wrote the Regulae. Between 1629 and 
1650 Descartes's letters give much information about 
his writings but say nothing of the Regulae. He is 
known to have spent the years 1618-25 travelling 
and soldiering; from 1625 to 1628 he stayed in Paris, 
which he found distracting and so unfavourable to 
study that he decided to go into retirement and, before 
he repaired to Holland 'to seek solitude' 2 (as we learn 
from one of his letters), he spent the winter in the 
country in France 'where he made his apprenticeship/ 3 
It seems from the context, however, that this means 
little more than that he accustomed himself to solitude 
at this time. Adam, nevertheless, suggests that it was 
then that the Regulae was written; but Gilson believes 
that it can be put much earlier and goes back to the 

1 Cp. A. & T., X. pp. 486-8. 

2 A. & T., X, p. 487. 

3 A. & T., V, p. 558. 

B *7 

time when Descartes was engaged on his first work, 
which was to have been called, Studium Bonae Mentis. 


In the A version the title given is 'Regulae ad Direc- 
tionem Ingenii' : in the H. Version 'Regulae ad Inquir- 
endam Veritatem'. Baillet combines these and Leibniz 
refers to The Search after Truth*. Perhaps the original 
MS. had both, either as alternatives or combined as 
Baillet has them. 


RULE i : The first of Descartes's rules is not quite as 
simple and obvious as it appears. The ultimate aim of 
study should be to guide the mind so that it can pass 
solid and true judgements on all that comes before it. 

In the exposition Descartes begins by contrasting art 
or craft with science or speculative knowledge. Art in- 
volves the acquisition of some special bodily skill, so 
no one person can be master of all the arts. Science 
needs no bodily training, no development of the body 
or any part of it. The power of knowing is a purely 
spiritual power, single, self-identical and absolute (as 
opposed to bodily skills). It retains its single character 
in every field. One may call it 'human or universal 
wisdom', or 'good sense* (bona mens), or the 'nat- 
ural light' of reason. It must be regarded as a spiritual 
light which is no more modified by the diversity of the 
objects it illuminates than is sunlight by the things on 
which it shines. It is an intellectual vision, a single, 
natural power of discriminating the true from the 

The knowledge of a science (unlike a craft) does 
not destroy, but increases the power to learn others. 


All sciences are interconnected, so that it is easier to 
learn all together than each in isolation. They are sim- 
ply the universal wisdom variously applied. If we want 
to search for the truth about things, therefore, we 
must make it our object to increase the natural light of 
reason in ourselves. The same vis cognoscens is at 
work throughout, and we must make it our whole aim 
to increase this perfectly general power. Indeed there 
are legitimate results to be had from the study of the 
special sciences (e.g. the cure of disease and the sup- 
reme pleasure of the vision of new truth); but these 
special rewards are irrelevant and to aim primarily at 
them would be to endanger the success of our main 
enterprise. We must think only of cultivating the nat- 
ural light. 

The last sentence of the exposition of Rule i 1 seems 
to insist on the supremacy of the practical end, but 
full and careful study shows that Descartes's real 
purpose was to demonstrate the identity of reason 
in speculation and in practice. The same bona mens 
is the condition both for the discovery of truth 
and for the conduct of life. Intelligent insight must 
precede judgement. All judgement whether specula- 
tive or practical is for Descartes (or so at least he says 
later) an act of will assent to or dissent from an idea 
which soon goes beyond all knowledge by mere 
observation. Thus it is better to pursue studies with a 
general aim than to work at special problems. 

We must attend to two matters in this exposition : 
(i) The severance of the power of knowing from all 
corporeal functions and (ii) its singleness, (i) The first 

1 A. & T., X, p. 361. 

is only lightly touched on here. In the exposition of 
Rule xii, 1 however, he supplements this and sketches 
in detail his view of the knowing subject, the doctrine 
underlying which seems to be the same as that elabor- 
ated in the Meditationes and the Prmcipia Thiloso- 
phiae (I, viii, et. seq). In Rule xii, however, he does 
not attempt to explain what the mind or the body is or 
how the body is informed by mind, but is content to 
put forward his theory of the knowing subject merely 
as an hypothesis, which may be summarized as fol- 
lows : 

There are in the knowing subject four faculties: 
sense, imagination, memory and intellect, of which 
only the last can perceive truth, the other three play- 
ing subsidiary parts. What in us knows is purely 
spiritual, not a bodily function nor conditioned by the 
body. This power may cooperate with and apply itself 
to sense, imagination and memory : it may attend to 
them; but the activity of knowing is attributed to a sin- 
gle spiritual power only the vis cognoscens, by which 
in the true sense, (proprie) we know. When its activ- 
ity is pure, then we are said intellegere, and the 
faculty concerned is intelligence. It is in consequence 
of this purely spiritual power alone that there is any 
knowledge (properly so-called) in human experience. 

Here, then, Descartes characterizes sense, imagin- 
ation and memory purely as the properties of bodily 
functions and changes. It is only as such that there is 
anything distinctive about them. True, these bodily 
functions and changes are connected with the activity 
of the spiritual faculty, which may attend to, and so 
make use of, them; but, strictly, sense, imagination 

2 Ibid., pp. 411-418. 


and memory are not different forms or grades of 
knowing. The knowing is always the same through- 
out and is due to a single spiritual principle a com- 
mon, abstractly identical element in them all the 
activity of the self-conscious or rational soul. To 
characterize the three subsidiary faculties, therefore, 
we must attend to what is peculiar to them and leave 
the intellect out. 

So considered, in their proper nature, they are cor- 
poreal organs, with definite location and extension. 
The action of other parts of the body produces in 
them, by the ordinary laws of extension and motion, 
physical changes : i.e. sensations, images, etc. 1 Descar- 
tes says, therefore, that we must regard sensation as a 
change in which we are passive. Our external parts 
sensate, strictly, only by being acted upon. No doubt, 
when we apply our peripheral sense organs to an 
object, it is an action which we initiate, but sensating 
itself is a purely physical change in the organ. It is a 
change of shape or form (idea) produced by the object 
on the surface of the organ, as real as the change of 
shape on the surface of wax produced by a signet. 
This change of shape is instantaneously communicated 
to the central part of the body, which Descartes (with 
the Schoolmen) calls sensus communis, the Common 
Sense, but this communication involves no material 
transference. 2 In principle it is the same as the move- 
ment of the pen as it writes: that of the pen-point 
being simultaneous with that of all parts of the pen 
and of the pen as a whole. 

The peripheral organ of sense communicates to the 

1 Cp. A. & T, X, pp. 412-13. 

2 Vide ibid., p. 414. 

central organ (sensus commums) changes ot shape or 
form (figura vel idea) and common sense stamps these 
on the phantasia or imagination just as a signet 
stamps shapes in the wax (and this, be it noted, is no 
mere analogy but a literal comparison). The phantasia 
is a real part of the body (vera pars corporis), which 
possesses a determinate size and is situated in the brain. 
It can assume different shapes in its several portions, 
so that it can hold and retain a plurality of distinct 
shapes (or ideas). So regarded it is memory. 

The way in which the spiritual power (intellectus) 
cooperates with these bodily organs is not, according 
to Descartes's account/ very clear. He says that either 
it receives a shape simultaneously from common sense 
and the imagination, or attends or applies itself to 
shapes preserved in the memory, or forms new shapes 
in the imagination. The vis cognoscens sometimes suf- 
fers and sometimes acts; is sometimes the signet and 
sometimes the wax, but here the simile is mere 
analogy. There is nothing in physical things in the 
least like the power of knowing. 

Hence the phantasia is a genuine part of the body 
and phantasmata are bodily changes in it. Descartes 
insists that these alterations involve no material trans- 
ference from the external thing to the sense-organ 
(e.g. physical particles), or from the sense-organ to the 
sensus communis and from it to the phantasia. Clearly 
he wishes to free himself from the confused concep- 
tions of curious entities (like species intentionales) and 
the use made of them by the theory of knowledge 
taught at the Jesuit School of La Fleche in which he 

1 Ibid., pp. 415-16. 

had been brought up. 1 He is also anxious to emphasize 
the contrast between the extended body and the 
purely spiritual intelligence of mind. Yet he does still 
retain material intermediaries between the mind which 
knows and the object perceived : forms or shapes in 
imagination. And these phantasmata are genuinely 
corporeal; yet they are somehow properties, or qualit- 
ies, not attached to any bodies or tied to any corporeal 
substance. They are shapes or changes which travel 
without being attached to any material particles (cp. 
the modern notion of 'waves'). 

This strange and obscure doctrine of the bodily 
phantasia is always in the background of the Regulae. 
(ii) The Singleness of the vis cognoscens. 

Why does Descartes emphasize the singleness of 
the vis cognoscens? Is it in order to suggest the 
mutual dependence of all truths and the unity of all 
sciences ? His language here and later suggests that the 
power of knowing is single in an abstract (or monoto- 
nous) sense. All details in any region of the knowable 
and the different regions of knowledge are one merely 
in the sense that they are all perceived by the one, 
single, undifferentiated power all bathed in the 
single, undifferentiated spiritual light. This seems to 
imply that the mutual interdependence of truths in a 
science leaves the truths themselves unaffected. By a 

1 Cp. Gilson. La Philosophic de Saint Bonaventure, pp. 146 ff., on 
the subject of phantasmata and the part they play in the 
theories of Aquinas and the Schoolmen; esp. pp. 158 ff. on 
species intentionales in the Summa Philosophica of Brother 
Eustace of St. Paul, a textbook which Descartes himself studied 
at school. In a letter written in 1640 (A & T, III, p. 185) he 
says that he remembers some of it. 

single science Descartes could mean an aggregate of 
unit truths in mutual isolation, not a whole system of 
truth. Similarly, if this is what he intends, the unity 
of the sciences seems only to mean that these collec- 
tions or complexes of unit truths may themselves be 
gathered into an aggregate of aggregates. We do not 
yet, however, know properly how Descartes does un- 
derstand this singleness of the vis cognoscens or unity 
of the sciences. 


RULE ii : We ought to study exclusively subjects 
which our mind seems competent to know with a cer- 
tainty beyond all doubt. 

RULE in : In whatever subject we thus propose we 
must enquire not what others believe but only what 
can be clearly perceived or with certainty inferred, 
these being the only ways in which genuine knowledge 
can be acquired. 

In the exposition of these two rules, there are two 
main points to which we must attend : (i) Descartes's 
attitude to Mathematics and (ii) his conception of the 
power of knowing as comprising both intuitus and 
deductio both insight and illation. 

The first will involve an account of Descartes's con- 
ception of mathesis vera (or universalis) and this 
may be postponed until we come to discuss Rule iv. 
But we may consider here why Descartes regarded 
mathematical science as the most perfect form of 
science, and what is its special propaedeutic power? 1 

1 Cp. Discours, A. & T., VI, pp. 19-22. 

Two things impressed Descartes in the contemporary 
'vulgar' arithmetic and geometry : (i) its infallible cer- 
tainty and self-evidence 1 about any matter there is 
one truth only, so that a child who has done an addi- 
tion sum has found out all that the mind of man could 
discover relative to it; (ii) the way in which this self- 
evidence expands to cover whole intricate problems. 
We frequently find a solution to a most complex prob- 
lem by long trains of reasoning in which each step is 
very simple yet quite infallible, and the steps are so 
ordered that we can go easily from one to another. 
Descartes was firmly convinced that knowledge, in the 
only proper sense (scientia) is certain, evident, indubi- 
table and infallible in sharp contrast with conjecture 
and opinion, however probable, or thinking which is 
susceptible of doubt in however small degree. 

Nor must knowledge be confused with memory. To 
remember is not, qua memory, the same thing as to 
know; not even the memory of demonstrations. We 
might remember all Euclid without knowing it : for 
knowledge is spiritual insight into the matters which 
may be marshalled by memory. 

On this view no science (except, perhaps, arithmetic 
and geometry) will stand the test, and only mathemat- 
ics will survive this definition of knowledge. All other 
sciences give conclusions which are doubtful, or even 
errors; mathematics alone contains truth and nothing 
but truth, free from falsity and doubt. How can this 
be ? Descartes early asked himself what gives absolute 
certainty to this science : why the power of knowing 
has only attained perfect realization here. And he con- 
cluded that it was due to the extreme purity and sim- 

1 Ibid., p. 21. 

plicity of the objects with which the geometer and the 
arithmetician are concerned. They presuppose nothing 
dependent on experience, nothing requiring confir- 
mation by experiment or observation. The data are 
entirely simple, abstract and precise; and these sciences 
consist in logical expansion of such data, rationally 
deducing consequences from them. 

Now Descartes maintains that the power of know- 
ing is neither more nor less than (i) the power of seeing 
simple data spiritual insight (intuitus) and (5i) the 
power of moving uninterruptedly from simple to sim- 
ple the power of illation (illatio). This continuous 
movement is such that all the links and every con- 
nexion are seen by the mind with the same immediate 
and infallible insight as that with which it intuits the 
data themselves. 

In arithmetic and geometry we have the only satis- 
factory realization of knowledge, precisely because 
they contain simple data such as it is the very nature of 
the intellect to perceive, and everything else is merely 
the formation of chains in which simple is linked to 
simple just such necessary expansion and connexion 
of data as it is the nature and function of the intellect, 
in its illative movement, to effect. 

So the essence of Descartes's method is (i) to admit 
no step which is not self-evident and (ii) in moving 
from step to step, to follow the inevitable logical order. 
This was the result of reflection upon the actual pro- 
cedure of geometry and arithmetic. He thought that 
these conditions would be fulfilled so long as the in- 
tellect worked according to its own proper nature. But 
he came immediately to the realization of defects in 
the existing mathematical sciences and thought a new 


universal mathematics to be necessary. This brought 
him to the conception of a possible universal and flaw- 
less mathematics. Tested by the two indispensable 
requirements, even Arithmetic and Geometry, as then 
taught, fell short of what he thought a perfect science 
ought to be. This explains the guarded language at the 
end of the exposition of Rule ii : * that we should not 
study Geometry and Arithmetic alone, except for 
disciplinary purposes (and even they would be better 
served by universal mathematics 2 ). This rule then as- 
serts only that we must not study anything which is 
not as certain as mathematics. 

Descartes's account of the Intellect 
(i) Intuitus. Descartes speaks at times as if intuitus 
and deductio were two quite distinct powers, faculties, 
or activities of the mind. It is, however, unlikely that 
he ever held so crude a view, or, if he did, he soon 
abandoned it. Nevertheless, he begins by characteriz- 
ing intuitus as a distinct act or function of mind direc- 
ted upon a distinct and special kind of object. It is 
intellectual 'seeing' and has a certainty peculiar to it- 
self, which 3 must not be confused with the vividness 
of sense-perception or imagination. 

As an act of mind intuitus is a function of the intel- 
lect expressing its own nature. Sometimes what we 
intuit is a material or corporeal thing, or a relation 
between such things. In this case, imagination will 
help, if we visualize the bodies; or sensation may 

1 A. & T., x, p. 366. 

2 Vide Rule IV. 

3 Cp. Reg. III. 

help, if imagination is directed upon the shapes in the 
sensus communis. Still, intellectual seeing must be 
clearly distinguished from sensation and imagination, 
and its certainty must be clearly distinguished from 
mere imaginative (or sensational) assurance. So Descar- 
tes begins by explaining what he does not mean by 

The intellectual certainty with which I see the mut- 
ual implication of self-consciousness and existence is 
immediate, like sense-perception; but, in the case of 
sense-perception, my assurance fluctuates. Sensation 
flickers and varies according to the illumination, or the 
state of my eyesight, or similar changing conditions. 
But the certainty of intellectual insight is steady, con- 
stant and absolute. To see a truth that x implies y 
is to see it absolutely and timelessly, once for all and 

The danger of confusing intuitus with imaginatio 
depends on the fact that, in both, two or more ele- 
ments are connected and the pictured union may be 
more vivid than the conceived 'cohesion'. But we 
must not confuse 'co-picturable' with 'co-thinkable' 
(conceivable) for these are not the same. Nor is 'un- 
picturable' the same as inconceivable. 

The text, however, is obscure. In Rule iii 1 he writes 
'Per intuitum intelligo non fluctuantem sensuum fid- 
em, vel male component's imaginationis judicium fal- 
lax' . And in Rule xii 2 he seems to imply that ideal 
elements may be composed in imagination in two 
quite different ways: (i) as dictated by the intellect 
and (ii) alogically and arbitrarily, due to the order of 

1 A. & T., x, p. 368. 

2 Ibid., pp. 416 and 421-4. 

succession or co-existence originally produced in the 
peripheral organ of sense. This second kind of combin- 
ation may be accepted and confirmed by the subject, 
to whom it may dictate a judgement, which is then 
likely to be erroneous (male componentis imagination- 
is) for which imagination alone is responsible and 
which is apt to deceive us. Connexion resulting from 
mere casual association is doubtful, for, unless con- 
trolled by the intellect, imagination often connects 
and compounds elements which do not really belong 
together and so should not be connected. Judging this 
connection to be fact, we are, consequently, in error. 
Descartes's mature theory is that judgement involves 
assent or dissent to a content conceived and this 
assent or dissent is an act of will. This theory is 
hinted at in the last sentence of the exposition of 
Rule i; but apart from this hint he seems, at this stage, 
to be working without any special theory and to be 
merely accepting the traditional scholastic view. In 
Rule iii, belief is seen to be an obscure topic and is 
s^id to be an action, not of the mind but of the will. 1 
At this date, therefore, Descartes presumably did not 
attribute to the will judgements in which we assent to 
rules we know or certainly apprehend. Likewise, in 
the exposition of Rule xii, 2 he distinguishes the faculty 
by which the intellect sees and knows (intuetur et 
cognoscit) from that which judges in affirming and 
negating, and here the second faculty is not identified 
with the will. 

Having shown what intuitus is not, Descartes goes 

1 Ibid., p. 370 : ' . . . fides, quaecumque est de obscuris, non 
ingenii actio sit, sed voluntatis.' 

2 A. & T., X, p. 420. 

on to state what it is. 1 It is a conception, formed by 
pure and attentive mind, so easy and distinct that no 
uncertainty remains. It is thus free from doubt, draw- 
ing its origin solely from the light of reason, and is 
more certain, because more simple, than deductio, 
which, however, even in men, is (on certain condi- 
tions) infallible. 

'Simplicity', here, does not mean that what is 
apprehended is atomic. Descartes speaks of 'simple 
natures' (naturae simp/ices), but refers to them also as 
propositions, and they are, in fact, always couples 
of terms in immediate logical relation. Each is genuine- 
ly simple : the object is apprehended, not discursively 
but in its entirety; and both the object and the act 
itself are present together, all at once and without 
lapse. In the exposition of Rule iii 2 one of the charac- 
teristics by which intuitus is opposed to deductio is 
praesens evidentia. This is essential to intuitus, but not 
to deductio, which can borrow its certainty from mem- 
ory. In the expositions of Rules xi and xii, the same 
point is brought out from the side of the object. In the 
first 3 it is stated that a proposition must be apprehen- 
ded (i) clearly and distinctly and (ii) all at once, not 
successively. And in the second 4 we are told that we 
must be in error if, in regard to a simple nature, we 
judge that we know it in part only. It is either wholly 
present and completely revealed or not at all; either it 
gives us the absolute and entire truth or no intellectual 
insight at all. 

1 ibid., p. 368. 

1 Ibid., p. 370. 
Ibid., p. 407. 
Ibid., p. 420. 


The origin of the whole doctrine is obviously Aris- 
totelian. Intuitus is the same as vowig, 'simple natures' 
are ra asrAa or ra a^tWera, and the truth of 
intuitus is, like the truth of v6r}<ri$ l opposed, not to 
error, but to blank ignorance. 

When Descartes says that the mind must be atten- 
tive and perception distinct, and that one of the two 
necessary conditions is that propositions should be 
apprehended clearly and distinctly, he is using more or 
less technical terms. So, in Principia Philosophic I, 
45, 2 to perceive anything clearly means that what 
is perceived is present and open to the mind attending 
to it, just as objects of sight are clear to the eye when 
they strike it 'satis lortiter et aperte'. To perceive 
distinctly means (in addition to perceiving clearly) 
that one has before one's mind precisely what is 
relevant, no more and no less. 

Therefore, in an act of intuitus the intellect alone 
must be engaged, and must be concentrated upon an 
object present to it in its single entirety an object 
which is openly and manifestly present to it. This sin- 
gle intentness must, moreover, include all that is essen- 
tial to the object (every relevant element in it) and 
must include nothing else. 

What is it that we see in such an act of intuitus? 
What is the character of the self-evident object of the 
intellect? Under Rule iii Descartes gives examples as 
follows : Thus anyone can perceive by the mind that 
he exists, that he is self-conscious, that a triangle is 
bounded by three lines only . . . ' 3 The object of intuitus 

1 Metaphysics Q t 1051^17 ff. 2 A. & T., VIII, p. 22. 

3 Ibid, p. 368 : 'Ita unusquisque animo potest intuere, se existere, 

se cogitate, triangulum terminad tribus lineis tantum . . ' 


therefore, is a proposition in which two elements are 
in immediate but necessary connection as implicans 
and implication. A immediately and necessarily in- 
volves B an immediate necessary nexus of a couple of 
elements. The nexus however need not be reciprocal, 
as Descartes specially tells us that it is not necessary 
that B should immediately implicate A. 1 He thus 
speaks of these self-evident data as propositiones or 
enunciationes* and he quotes, as further examples, 
'2 + 2 = 4', '34-1=4', '2 + 2 = 3+ i'. 3 Yet he con- 
stantly speaks of these self-evident objects of intuitus 
in terms which suggest concepta an 'A' or a 'B' and 
not a complex 'A implying B'. He speaks of them as 
res simplicissimae and naturae purae et simplices. 

It is not easy, at first sight, to reconcile such passages 
with what must be taken as his considered view : that 
the object is a nexus. There is a very puzzling passage 
in the exposition of Rule xii, 4 which may be summar- 
ized as follows : 

We must distinguish what is a single thing, when 
things are considered per se, from what is single when 
considered from the point of view (or as an object) of 
our thought. E.g. a shaped body is a single thing when 
considered 'ex parte rei', but as an object of our know- 
ledge it is complex it is compounded of three 'nat- 
urae' : body, extension and figure. Though these 
have never existed apart, they must be conceived sep- 

1 Ibid., p. 422. 

2 Ibid., pp. 369, 370; 379; 383. 

3 Ibid., p. 369. 

4 Ibid., pp. 418-425. Note : The doctrine is repeated in various 
passages of Locke's Essay Concerning the Human Understand- 
ing. Locke lived in Holland from 1683 to 1689. 

C 33 

arately before we can say that they are found together 
in one thing. In the Regulae we are concerned with 
things qua objects of thought. Hence 'a single thing' 
must mean what is so clearly and distinctly conceived 
that we cannot divide it: e.g. duration, extension, 
figure, motion, etc.; i.e. it is a not further analysable 
object of knowledge and thought. And it must be a 
genuine element of an object of knowledge, not a mere 
generality or abstract universal. 1 For instance 'limit' 
(in 'Figure is the limit of extension'), is more general 
than 'figure' (for there may also be a limit of duration 
or motion), but it is not more simple. It is complex, 
being a conflation from many simple natures, com- 
pounded of several different ideas by disregarding 
their differences. It is applicable to all only equivoc- 
ally, and not applicable, in any definite sense, to any. 
Descartes shows that all such simple objects of 
thought fall into three classes: purely intellectual, 
purely material or corporeal, and those which are 
common to both. By the first he means objects of 
thought which are intelligible to self-conscious beings 
or spirits only, and which are known without any 
image or other corporeal aid (e.g. knowledge, ignor- 
ance, doubt, volition, and what these are). The second 
are known to exist only in bodies, and intellectual 
insight into them is facilitated by imagination or sen- 
suous presentation (e.g. figure, extension, motion). The 
third class of objects is common because they are 
attributable both to material bodies and to spirits 
indifferently (e.g. existence, duration, unity). We 
must here include those common notions which are 
links connecting other simple natures and on the self- 

1 Cp. pp. 418-9. 

evidence of which the conclusions of our demonstra- 
tions rest (e.g. identity, equality, etc.: two things 
equal to the same thing are equal to one another). 
These common objects may be known by inspection of 
the intellect, pure and alone, or so far as seeing images 
of material things reveals them to the intellect, The 
list of simple natures must be extended to include the 
corresponding negations and privations of such con- 
cepts, so far as they are conceived by the mind : e.g. 
instant (the negation of duration), rest (the privation 
of motion), nothing (the negation of existence) and so 

This and similar passages suggest a sharp distinction 
between intuitus and deductio (common in phil- 
osophy), and the suggestion is confirmed in the next 
paragraph, 1 where he distinguishes two faculties of 
intellect, one which sees and knows, and one which 
judges by affirming and negating. There are certain 
symbols which form, so to speak, the letters of the 
alphabet of reality universals pervading either all 
things, spiritual and corporeal, or large areas of the 
real, but, nevertheless, in some sense singular and sim- 
ple, They are fundamental constituents of all that 
exists, and they themselves, though they do not (in the 
same sense) exist, yet they subsist. They are in some 
sense there, confronting the mind, waiting for its 
recognition which is direct and immediate a simple 
act of seeing. This is the only genuine knowledge, the 
only real and absolutely certain grasp of truth, the in- 
dispensable precondition of judgement and reasoning, 
which by combining and arranging what we intuit 
gives a more precarious and derivative knowledge. 

1 Ibid., p. 420. 


Yet any such interpretation is not only in direct 
conflict with what Descartes has said under the ear- 
lier Rules about simple propositions being the object of 
intuitus, is not only in conflict with what he says 
about deductio, but can hardly stand and does not, on 
closer examination, really emerge from the passage 
under Rule xii, especially if we consider Descartes's 
examples and his manner of expressing himself. 

He says that no corporeal idea can be imagined such 
as to represent to us what doubt, knowledge, etc., 
really are. It seems, then, to be implied that what we 
intuitively perceive is 'that knowledge is so and so*. 
Again, speaking of the second class of simples, he 
says that they are known to be only in bodies (that 
duration is attributed to certain bodies, that 'this body 
is in motion', etc.); and, of the third, that they are 
attributed indifferently, now to spirits, now to cor- 
poreal things (e.g. 'that this mind exists', 'that this 
body is extended at rest, etc.'). There seems no room 
for hesitation if we bear in mind that Descartes adds to 
the list of simple natures the principles of linkage of 
our knowledge, the universally accepted laws of 
thought. What the intellect perceives, then, is quite 
clearly, at least two elements in one fact, two ele- 
ments in immediate and necessary cohesion. The 
whole fact, in necessary combination, implicans plus 
implicandum is 'simple' in that it is a minimum of 
knowledge. Nothing less is knowable at all. We can- 
not know 'A' nor 'B', nor 'implying' except in a 
single unitary whole where all three are distinct (no 


doubt) but inseparable. If we know at all it must be 
'A implying B'. 

(2) Deductio. The first mention of deductio is under 
Rule vi, where it is contrasted with experience (empir- 
ical observation), and these (deductio and 'exper- 
ience') are regarded as two alternative ways by which 
we can arrive at a knowledge of things. Assuming that 
we begin with a self-evident truth, we may extend our 
knowledge by rationally following the implications 
of the implicans the illation of one thing from an- 
other. Experience often misleads us, but deductio is 
quite as infallible as intuitus itself. We may fail to 
make a deduction, we may fail to draw out what is 
logically implied, but no intellect at all rational can 
mis-infer or mis-think any more than it can mis-per- 
ceive. That is why mathematics is free from error; it 
is no more than the following out of the logical im- 
plications of simple, abstract and absolutely certain 
data. These, if they are perceived at all, must be per- 
ceived as they are, and their implications, as logically 
drawn out, must be infallible. Observation, on the 
other hand is fallible. 

So far deductio takes its place alongside intuitus as 
one of the two necessary and only acts of the vis 
cognoscens. It is the very nature of the intellect to 
perform both of these activities, and thus we achieve 
knowledge without any deception. To understand is 
intellectually to perceive or to deduce or both, and 
intellectually to perceive or deduce is to understand. 
There is no such thing as false intuition or faulty 
deduction. So it appears that one of these native func- 


dons of the mind assures us of the data and the other 
guarantees our advance from the data. 1 

It cannot be denied that the interpretation here 
given is what Descartes intends. More than once in the 
Regulae he asserts the duality of original acts of the 
intellect and distinguishes intellectual vision from il- 
lative or discursive movement, and the perception of 
simple reals from the linkage or connexion between 
simple reals. But in spite of this explicit doctrine, there 
runs through the Regulae a more adequate conception 
of the function of the intellect, though one incompati- 
ble with the doctrine expressed. This more adequate 
doctrine, which first appears under Rule iii (just as 
Descartes is formulating the above, more crude, 
theory), is the more important for our present pur- 
pose. 2 We need not discuss whether Descartes ever 
was a whole-hearted believer in the mechanical 
analysis of the power of knowing into two functions 
and the corresponding division of the objects of 
mind into simple natures and the linkages between 
them. May he not simply have adopted an expository 
dfevice ? If we think that he really believed it, we must 
suppose that he was later forced beyond it, but that he 
never became fully conscious of his own advance and 

1 In the passage under Rule III (A. & T., X, p. 368) : omnes 
intellectus nostri actiones, per quas ad rerum cognitionem 
absque ullo deceptionis metu possimus pervenire : admittuntur- 
que tantum duae, intuitus, scilicet et inductio' the word 'in- 
ductio' is probably an error in the MS., for under Rule IV 
(ibid., p. 372), Descartes refers again to this passage using the 
word 'deductio' (cp. also Reg. ix Exp., ibid., p. 400). The matter, 
however, is complicated by the fact that what Descartes means 
by 'enumeratio sive inductio, is uncertain (cp. below, pp. 49-61). 

2 Cp. p. 25 above. 


so fell into verbal self-contradictions. On the assump- 
tion that he did not really hold such a view, we must 
suppose that his contradictions are corrections of what 
he considers to be an inadequate way of expressing his 

Under Rule iii, Descartes proceeds to say : 'But this 
self-evidence and certainty of intellectual insight is 
required not only for simple propositions but also for 
all discursive reasoning. For example, 2 and 2 yield the 
same sum as 3 and i, in this case we must perceive not 
only that 2 and 2 make 4, and 3 and i likewise make 
4, but also that the conclusion is a necessary conse- 
quence of these two propositions'. 1 

Under Rule xi we get the same idea: The simple 
deduction of any one thing from another is effected by 
means of intellectual insight. 2 And under Rule xii : 
'. . . the mind's insight extends to the apprehension 
of simple propositions, their necessary linkages and 
everything else which the intellect experiences with 
precision'. 3 Thus intuitus is needed for both linkages 
and simple natures, and there is no sharp division be- 
tween the objects of intuitus and of deductio. The 
difficulty now is to distinguish deductio as an original 
act or function of the intellect as a separate mode 
of knowledge at all. How is it 'other than' intuitus, 
as is alleged under Rule iii? 4 

1 A. & T., X, p. 369. 2 Ibid., p. 407. 

3 Ibid., p. 425 : 'Atque perspicuum est, intuitum mentis, turn ad 
Illas omnes extendi, turn ad necessarias iUarum inter se con- 
nexiones cognoscendas, turn denique ad relinqua omnia quae 
intellectus practise, vel in se ipso, vel in phantasia esse ex- 

4 Ibid., pp. 369-70. 


In order to bring out Descartes's better view we may 
begin by giving the answer which he ought to have 
made; then we shall outline the position to which he 
seems on the whole to have inclined (apart from 
waverings and contradictions), pointing out where 
he explicitly departs from it; and, thirdly, we shall 
attempt to confirm this interpretation by detailed 

(a) Descartes ought to have said that two things are 
always essentially involved in every act of knowing : 
(i) a certain illative movement or discursus an in- 
tellectual analysis and synthesis in one which brings 
to light distinguishable elements and at the same time 
points to the logical implication by which each leads 
to the next the necessary connections by which they 
cohere; (ii) a certain unitary apprehension, an immed- 
iate, direct perception of the distinguishable elements 
(as opposed to isolable constituents) as indivisibly con- 
stituting a whole. 

Beyond this no further distinction of intuitus from 
deductfo is possible or necessary. In principle both 
modes are essential and indispensable to any and every 
act of knowing. And in principle the character of every 
cognoscibile and every cognitum must be such as to 
answer to these two modes of apprehension. Nothing 
is or can be known unless it is unitary and whole and 
present before the mind in its wholeness, and unless 
within this unity there are two or more distinguish- 
ables, so that it is discovered to the mind by a discur- 
sive movement at once analytic and synthetic. The 
distinguishables are seen in this discursus to be mut- 
ually implied, as it moves from one to the other along 
the line of logical connexion. The discursus is construc- 

tive of a whole and is so far synthetic, but we must 
not forget its analytic obverse, or we shall tend to 
attribute to immediate apprehension alone what re- 
quires a discursus as well, and to postulate the unit- 
ary perception of atomic cognoscibilia. 

The simplest act of mind is deductio and intuitus in 
one. The minimum cognoscible is a 'simple nature' 
which is also a proposition both conceptum and 
deductum : each of these only because and in so far 
as it is the other. Similarly, the most intricate piece of 
reasoning, or the most complicated system of demon- 
stration if in it we achieve genuine knowledge, is 
illative movement and direct intellectual vision in in- 
separable unity conceptum as well as deductum or 
demonstratum. If on the side of the intellect either of 
these is wanting, then the knowing in question is de- 
fective the knowledge is imperfect and the object 
incompletely intelligible and only partially understood. 
Owing to the weakness of the human intellect such 
maimed and limping efforts are at times accepted as 
genuine knowledge, and similarly mutilated objects as 
genuine cognoscibilia; and so we are led wrongly to 
assume, on the one hand, atomic elements which can 
be apprehended immediately and alone, or, on the 
other hand, long chains of reasoning which subsist 
without any real unity or wholeness; and to believe 
that the mind, in knowing thus piecemeal, knows a 
genuine object in a genuine manner. 

(b) What Descartes actually tends to maintain is a 
compromise between the above position and a neat but 
wooden analysis, with a sharp distinction between 
intuitus and deductio. (i) There are certain primary 
basal facts, reals or truths the letters of the alphabet 

of reality - knowable and known by the immediate 
intellectual apprehension of intuitus alone. These are 
at once elementary constituents of reality and the 
foundations of all knowledge, (ii) At the other extreme 
there are certain remote consequences of these data 
connected to them by chains of reasoning effected by 
illative, discursive activity only. The apprehension 
here takes the form of construction (or re-construction) 
of the chain of reasoning, enabling us, not to see, but 
to infer the consequences from the primary data, (iii) 
Between these two there is an intermediate region of 
knowledge, where what is known is both seen immed- 
iately (intuitively) and apprehended as demonstrable, 
or inferentially and necessarily deducible, from ulti- 
mate cognoscibilia. 

Nevertheless, Descartes only maintains this com- 
promise with qualifications. First, some of his state- 
ments show that he waveringly recognizes that the 
apprehension of primary reals and truths involves 
discursus and, secondly, the absence of immediate per- 
ception in long chains of reasoning is sometimes 
ascribed to an infirmity in the human mind. 1 What he 
says implies the recognition that so far as intuitus is 
absent our knowledge is neither genuine nor perfect. 

(c) Let us now consider what Descartes actually says 
in more detail, (i) First, he states a kind of compromise 
doctrine : some propositions may be said to be known 
at times by intuitus, or at others by deductio, accord- 
ing to different points of view, but first principles are 
known only by intuitus and remote conclusions only 
by deductio. 2 It is only such as can be immediately 

1 Cp. below on 'enumeratlo sive inductio', pp. 49ff. 

2 A. & T., X, p. 370. 

deduced from first principles to which the comprom- 
ise doctrine applies. Here the discursus may subserve 
perception, or perception assist the discursus. 1 Again, 
he says that many things, though not self-evident, may 
yet be known with certainty, provided only that they 
are deduced by a continuous and uninterrupted move- 
ment of thought from premises which are certain, and 
that the thinker perceives clearly each single step. 
Though we cannot embrace all the links of the chain 
in one act of perception, we can apprehend the con- 
nexion of the last stage with the first, without, in the 
same act, perceiving each several link, provided that 
we have seen all the links and their several connexions 
and remember that each was necessarily connected 
with its neighbour. So deductio is contrasted with 
intuitus, first, as movement or succession, and secondly 
so far as its certainty does not require the 'praesens 
evidentia' required by intuitus, but is in a manner 
borrowed from the memory. 2 The middle region of 
any science or department of the knowable is, then, 
from different points of view, the object both of 
intuitus and of deductio. 

Yet Descartes maintains a sharp distinction between 
deductio and intuitus, even where they co-operate. 
Both may be indispensable for a genuine act of know- 
ledge, yet within that act each remains detachable; 
each remains what it was when it constituted imper- 
fect knowledge by itself. In fact each always remains 
itself and what it is in isolation, even when both go 
together in one piece of knowledge. When Descartes 
says, or implies, that immediate apprehension and 

1 Cp. ibid., pp. 407-8. 

2 Ibid., p. 370. 


illation are inseparable and indispensable conditions of 
any full act of knowledge, he still views them as 
connected ab extra with each other. Each requires the 
other but is not fused with it. Thus, in order to know 
that 2 + 2 = 3+1 is a necessary consequence of the 
fact that 2 + 2 = 4 and 3+1=4, the intellect must 
perceive by intuitus that 2 + 2 = 4 an ^ 3 + i=4> then 
it must deduce the equality of 2 + 2 and 3 + 1, and 
finally it must (again) perceive that the conclusion is 
a necessary consequence of the premises. 

Descartes always tends to conceive reasoning as a 
chain of links or sequence of states a movement of 
thought along a chain of truths, each link being self- 
evident and the movement from link to link or rather 
the connexion of the second link to the first, after the 
movement has been made must be self-evident. From 
this point of view, Descartes's deductio is the same as 
the ideally perfect syllogism or owobttfyg of Aris- 
totle. It is true that he protests against syllogism, 
but unless he mistakes Aristotle he means by that the 
traditional subsumptive syllogism of the Schoolmen. 
With that he will have nothing to do, nor with their 
'dialectical 1 reasoning. But his own deductio is, never- 
theless, the same as Aristotle's 'complete demonstra- 
tion 1 . For the perfect asro5g//, 'A must be Y', has to 
resolve the interval between A and Y into a succession 
of minimal, self-evident steps; 'A must be B; B must be 
C, etc., so that A will be seen to involve B, C, D, and 
so on, leaving nowhere any interval without imme- 
diate judgement. Thus, between A and Y a distance 
will have to be traversed which, though completely 
and exclusively covered by self-evident steps, is itself 
too great for the connexion between the remote ex- 


tremes to be self-evident. This is the same as Des- 
cartes's doctrine. 

(ii) Nevertheless, even the apprehension of a prim- 
ary truth or self-evident simple nature (as we have 
already asserted) does involve discursus. This becomes 
apparent in Descartes's reply to the second set of 
objections to the Meditations. 1 The objectors urge 
against Descartes that he asserts that nothing can be 
known with absolute certainty unless and until the ex- 
istence of God the perfect, omnipotent and truthful 
being is known; yet he claims in the second Medi- 
tation, though still as yet uncertain of God's existence, 
clear, distinct and indubitable knowledge of his own 
existence; and afterwards asserts that this must be 
deduced from the knowledge of God's existence. Des- 
cartes replies 2 that 'cogito ergo sum' is a prima 
quaedam notio not deduced by syllogistic reasoning, 
yet at the same time he makes it clear that it is a 
nexus in which two factors, implicans and implication, 
are necessarily involved. (He means by 'syllogismus', 
here, the bringing under a universal rule of a particular 
instance 'syllogism' as understood by the School- 
men, but what Aristotle in the Posterior Analytics 
refuses to recognize as true syllogism and regards only 
as making explicit what is already implicitly known). 
Descartes easily shows that my own existence, so far 
from being deduced from a major premise, is prior to 
any universal major and I can only arrive at the certain 
knowledge of it by consciousness of what my own 
experience implies. Spinoza, in his summary of Descar- 

1 A. & T., vii, pp. 124-5. 

2 Ibid., p. 140. 


tes's philosophical principles/ follows this statement 
of Descartes's and rightly lays it down that 'cogito 
ergo sum' is not a syllogism; but when Spinoza goes on 
to summarise the position, saying that 'ego sum cogit- 
ans' is a single proposition equivalent to 'cogito ergo 
sum', he has overshot the mark. For Descartes says 2 
that a man learns the universal premise from the fact 
that he experiences in his own case the impossibility 
of thinking unless he exists. This experience is said to 
be a recognition by the simple insight of the mind, 
which seems very forcibly to emphasize that what the 
mind intuites is a necessary nexus or implication. 
Thus 'cogito ergo sum' is, as Spinoza says, a single, 
unitary proposition (unica propositio). But Descartes 
himself shows that it includes within itself an illation 
from one element to another. It is a mediate judge- 
ment, a concentrated or telescoped inference. The self- 
evident fact ('res per se nota') is 'that my thinking 
necessarily implies my existence'. 

We may compare this with what Descartes says in 
the exposition of Rule xii. 3 He has just been distingui- 
shing three classes of simple natures, all of which 4 
(e.g. figure, extension, motion, and the like) are 
res per se notae. But he proceeds to say that they are 
conjoined or compounded with one another and that 
this conjunction is either necessary or contingent. It is 
necessary when one is confusedly implied in the 

1 Renati des Cartes Trincipia Philosophiae, more geometrico 
demonstrate, I, Prolegomenon (Opera, Vol. IV, Van Vloten en 
Land, pp. 112-3). 

2 Loc. cit. 

3 A. & T., X, p. 42i. 

4 vide, p. 420. 

lotion of the other so that we cannot conceive either 
properly (if at all) if we try to conceive them as inde- 
pendent of each other: e.g. extension and figure or 
motion and time. Likewise 4 + 3 = 7 is a necessary 
composition because the notion of seven must con- 
fusedly include those of four and three. 

The first part of this passage, in spite of its obvious 
formal inconsistency, must be treated with all respect 
because it contains the germ of an important later 
theory of Descartes. The formal inconsistency is that 
he says motion, extension, etc., are simple natures 
per se notae out of which all our succeeding knowledge 
is compounded; yet here motion, duration and figure 
have become implicated in larger and more concrete 
concepts, and it is these which are clear and distinct, 
per se notae, and knowledge of these is the necessary 
precondition to that of the so-called simple natures. 
This inconsistency, however, is of minor importance 
What must be noted is that the line of thought here 
implied would lead to the recognition of two clear 
and distinct, or self-evident, natures only: substantia 
extensa and substantia cogitans there would be a 
necessary system of extended implicantia and impli- 
cata and also one of spiritual implicantia and impli- 
cata. Here the Cartesian conception is adumbrated of 
a physical universe, open to thought as a coherent 
system of mutually implicated data (motion exten- 
sion, figure, etc.), and its correlate, a corresponding, 
self-contained, coherent system of spiritual concepts. 
There is both external and internal logical coherence, 
and the physical universe is transparent to intellectual 
insight because and in so far as thought distinguishes 
certain characters, and in recognizing them is driven 


to illate from one to another and see them as elements 
of a self-integrating whole. 

But what are we to make of Descartes's statement 
about 4 + 3 = 7? We saw, in the explanation of Rule 
iii, 1 that 2 + 2 = 4 and 3 + 1=4 were cited as examples 
of objects of intuitus, and intuitus alone. They were 
facts which our intellect alone could directly see, no 
other method of cognition being required. We should 
expect, therefore, that 3 + 4 = 7 would be the same. 
But here it is called a composition of simple natures, 
and one which is necessary and not contingent be- 
cause the conception of 7 involves confusedly 3 and 4. 
What then are the 'simple natures'? 3 and 4, or 3 
+ 4, or 7 ? It seems that 3 and 4 could not be simple if 
7 is not. Yet the conception of 7 is clearly admitted to 
include the conceptions of 3 and 4 as well as their ad- 
dition. It seems that here Descartes, consciously or 
not, does recognize a movement of thought within 
and constituting the clear and distinct intellectual 

So far as any reasoning remains sheerly discursive, 
without there being any comprehensive conception, 
or immediate intuition covering the whole, such 
reasoning is not itself knowledge but an imperfect 
substitute for knowledge a limitation or stunting of 
knowledge which the mind accepts only because of 
its infirmity. This also is implied at times by Descar- 
tes's express assertion, but to decide how far he inten- 
ded it we should have to be clear about the meaning 
of enumeratio sive inducio', which is very difficult to 
determine and can be assigned only conjecturally. 

1 ibid., p. 369. 

(3) Enumeratio sive inductio. 

Descartes holds that every possible subject of inquiry 
is in principle intelligible that is, reality as a whole 
and every department of the real, every one of its 
parts, is a system of intelligibles necessarily connected 
in inherent logical order. This logical order of linkages 
is followed by the intellect in knowing, because and in 
so far as it is neither more nor less than the order 
proper to intuitus and deductio. Intuitus reveals the 
first link in the chain, and provided we advance from 
link to link by correct logical illation, never breaking 
the continuity, then every element of every part of 
reality, as well as the whole, will, in due course be 
reached. In principle, then, careful logical analysis will 
enable us to find the links in any sphere of inquiry, 
so long as we adopt the proper logical order. Every- 
thing can be known in this way with the same certain- 
ty and by the same means. Nothing is too remote; 
everything can be reached with the same ease, and the 
knowledge of one thing is never more obscure than 
that of another, for all complexity consists in "rerum 
per se notarum compositio' (cp. Rule xii, Expos.). 

Reality, as a whole is, accordingly, perfectly intel- 
ligible, and so is every group of facts; all are, and are 
knowable as, self-evident constituents self-evidently 
ijiter-connected. That is so, at least in principle and 
ideally i.e. it would be so for a mind which had 
mastered every department of knowledge and expoun- 
ded or unfolded it all in the correct logical order as a 
single chain of self-evident truths. Descartes, however, 
seems, at least from Rule v onwards, to conceive ideal- 
ly perfect knowledge as a network or system of chains, 

D 49 

rather than as a single chain; and, secondly, he regards 
each link as less complex and more simple than its 
successor more complex and less simple than its 
predecessor. But no human mind is capable of master- 
ing every subject of inquiry and all knowledge, so the 
sum total of our knowledge could not unroll itself (in 
fact) in a chain, or system of chains, self-evident in 
every link. 

If we reflect upon man's efforts to systematize his 
knowledge, we shall notice the following three typical 
cases : - 

(i) A single line of logical implication lies straight 
ahead of us, straight from what lies immediately before 
us to the conclusion or solution of our problem. In this 
most favourable case, the whole problem the whole 
segment of the real which is being studied has resolv- 
ed itself, bit by bit, into its ultimate self-evident con- 
stituents and their logically inevitable and self-evident 
linkages. The only difficulty here arises from the 
growing length of the chain. When we try to set out 
the connexions in the best logical order to connect 
all the simples discovered, and arrange them as a con- 
tinuous inevitable illation, with every pair of simples, 
from first to last, self-evidently connected we some- 
times get a chain so long that we cannot keep it, all at 
once and as a whole, within the grasp of our intellec^ 
tual vision. We shall thus be forced to rely to some 
extent on our memory, and that is fallible. Strictly 
speaking we do not genuinely know the connexion 
of the last element with the first unless we can demon- 
strate it by an unbroken, continuous movement of 
clear thought. This is never possible if any one link is 

lost or misplaced, which is liable to happen when we 
rely on memory; and so is hardly ever possible. 

The remedy for this, described by Descartes in Rule 
vii, is enumeratio : the repeated reviewing of every 
link in the long chain, and re-thinking of the various 
proofs. Such practice facilitates and strengthens our 
powers of illation and brings longer and longer chains 
within the span of intuition. By this means we may 
extend our powers of deduction to any length, pro- 
gressively reducing our reliance on memory. Descartes 
never suggests that the role of mere memory can ever 
be entirely eliminated in this way, but we can shorten 
the time required for the illation so that less is left for 
the memory to do and we seem to see all at once. 1 
So we may approach the single instantaneous intuition, 
and Descartes clearly assumes that our knowledge is 
the more perfect in the degree to which the discur- 
sive movement falls within the span of such a single, 
immediate intuition. 

To achieve complete knowledge of any subject we 
must survey (perlustrare), by a continuous and un- 
broken movement of thought, all matters which 
belong to our inquiry, singly and together, in a suffic- 
ient and well-ordered enumeration. 2 But what this 
'enumeratio' is he does not tell us precisely, for the 

^Cf. ibid., p. 388: '. . . donee a prima ad ultimam tarn celeriter 
transire didicerim, ut fere nullas memoriae partes relinquendo, 
rem totarn simul videar intueri'. 

2 Ibid., p. 387, Reg. vii. Cp. Discours, A. & T. VI, p. 19 : '. . . de 
faire partout des denombremens si entiers, et des reveues si 
generates, que je fusse assure de ne rien omettre'. Also, p. 550: 
'ut turn in quaerendis mediis, turn in difncultatum partibus per- 
currendis, tarn perfecte singula enumerarem et ad omnia cir- 
cumspicerem, ut nihil a me omitti essem certus'. 

enumeration he goes on to describe 1 has no relevance 
to the difficulty of the length of the chain (as opposed 
to the heterogeneity or complexity of the elements). 
It seems as if what he is advocating is no more than 
some special rubrics for grouping the details the 
special proofs or parts of the chain in fact, any kind 
of mechanical device to help the memory. 

(ii) But as a rule we are in a far less favourable 
situation, for even if we are following a single line of 
direct logical implication from a simple datum, our 
progress may be barred by the nature of the facts 
(the obstinacy of nature) or by our own relative ignor- 
ance and inability. We are trying to resolve a particul- 
ar problem of a special science; but, in the first place, 
in defining and delimiting the several domains of the 
different sciences, we do not divide reality into really 
separate, self contained and independent worlds, so the 
direct line of investigation may cut across the accepted 
boundary marking off the domain of our special 
science from those of other sciences. The mathe- 
matician, for instance, proceeds along a chain of 
mathematical reasoning, but he may find further pro- 
gress arrested by an inescapable barrier, if he comes 
upon a link which has no successor, or direct Implica- 
tum, within the domain of mathematics. The con- 
nexion would then, of necessity, be obscure to him as 
a mere mathematician, for the implicatum may belong 
to the domain of physics or optics. And, secondly, 
even if we could assume that the domain of each 
science is in fact self-contained and that the facts are 
linked in a single chain, still we do not know the 
whole science, but only work at a particular problem. 
1 A. & T., x, p. 388. 

Thus it would be very unlikely that the links we know 
should all be adjacent or neighbouring links in the 
chain, or that our analysis would unfold all the links 
in all the parts. Consequently, we should from time to 
time come upon a link of which the immediate neces- 
sary implication is not obvious to us and will not 
become so until we have learned more. 

(iii) But in many of our actual scientific reasonings 
no single, direct line of implication exists, or the 
manner of our investigation tends to conceal it if it 
does. For we often move, not from simple to simple, 
but from many elements taken together to a simgle 
consequent. The thread of implication is often twisted 
out of many strands. We must deduce our consequent 
from an antecedent in fact composed from a set of 
co-ordinate links or elements each drawn from a 
different implicatory sequence. 1 We might have said 
'induce', rather than 'deduce', for this process is 
generally called 'induction'. In such cases advance is 
obstructed, not only by the growing length of the 
chain, but also by the complexity and heterogeniety 
of the data and the intricacy of their interconnection. 
Here, according to Descartes, enumeratio is essential 
as an auxiliary of the deductive process. Indeed he is 
so impressed by its importance, that at times he 
speaks of it as though it were an independent method 
of proof and at others as if it were the only method of 
proof, as opposed to direct intellectual insight. 

Now 'enumeratio sive inductio' is said to be illation 
or inference derived and composed from many dis- 
connected things (ex multis et disjunctis rebus collec- 

1 Cp. A. & T., X, p. 429, 11. 19-27. 


ta'). 1 In a passage describing the same logical proced- 
ure, he speaks of 'proving by enumeration* and 'a 
conclusion drawn by induction'. 2 Yet in at least two 
other passages he treats of 'enumeratio* as the only 
form of deductio genuinely distinct from 3 and in no 
way reducible to intuitus* Here Descartes says : This 
is an opportunity to explain more clearly what was 
earlier said about intellectual insight: since in one 
place we contrasted it with deduction and in another 
with enumeration only ...' (i.e. not with deduction in 
general but with enumeration in particular). Simple 
deduction of one thing from another takes place 
through intuition (per intuitum), and for this two 
conditions must be satisfied : The proposition must be 
apprehended (a) clearly and distinctly, and (b) all 
together, not part by part. But, a deduction, when 
considered as about to be made, does not seem to take 
place all together. According to the exposition of Rule 
iii, deductio is a movement of the mind, inferring one 
thing from another. So deductio is rightly distinguish- 
ed from and contrasted with intuitus. But, if we 
consider a deduction as already made, it is not a 
movement but the termination of a movement ('nul 
lum motum . . . sed terminum motus'*). Therefore, 
when the content deduced is simple and manifest, we 
suppose it to be seen by intuitus. When it is complex 
and involved, we take a different view, and give the 
process the name 'enumeratio' or 'inductio', because 

1 Reg. xi, A. & T., X, p. 407. Cp. Reg. vii, p. 389. 

2 Ibid., p. 390. 

3 Ibid., p. 389. 

* Ibid., pp. 407-8. 
5 Ibid., p. 408. 

the conclusion cannot be apprehended as a whole, and 
its certainty depends in part upon memory (i.e. it 
depends on judgements specified in many and various 
parts of the proof). 

Yet, after all, if these passages only are considered, 
Descartes has shown no more than that, in complicat- 
ed processes of reasoning, enumeratio sive inductio is 
a useful, indeed an indispensable, aid to proof. Induc- 
tion is useful where the chain of reasoning is long, and 
in complex arguments where we have to lean more 
upon memory because the steps are too heterogenous 
to be perceived as a whole. A methodical grouping of 
these data and the numbering of the steps of the com- 
pleted argument are necessary to prevent them from 
slipping from the memory or becoming disarranged. So 
far, then, enumeratio is not a mode of proof at all. Its 
function is to arrange and group premises already 
intuited and steps of an argument already deduced, in 
order to help retention in the memory. It is not a 
method by which to acquire fresh premises or to infer 
anything fresh from premises we already have. 

But some other passages in the Regulae modify this 
position. In the exposition of Rule vii he says that this 
enumeration, or induction, is a scrutiny (perquisitio 
a Baconian term) of everything relevant to the prob- 
lem before us, careful and accurate enough for us to 
know that we have left out nothing of importance 
which ought to have been included; so that even if we 
fail to solve our problem we have advanced our 
knowledge at least to the extent that we perceive that 
the object for which we sought could not have been 
discovered by any method known to us. If we have 


surveyed all methods open to man, we may say that 
such knowledge is beyond the human mind. 

This explains why Descartes often uses enumeratio 
as a preliminary survey of the ground before he at- 
tempts to solve a problem. For instance in the exposi- 
tions of Rules viii and xii - we are given a survey of the 
implications of knowledge in regard to both possible 
objects of knowledge and possible instruments or 
powers of knowing. Both these passages are prelimin- 
ary to an attempt to show the limits and range of the 
human intellect. Similarly in the exposition of Rule 
vii 2 the example is given of surveys preliminary to the 
proofs that the area of a circle is greater than that of 
any other figure of equal periphery, and that the 
rational soul is not corporeal. 

Again in the Treatise on Dioptrics, 3 Descartes is 
about to explain the means of perfecting human 
vision : to determine what kinds, shapes and arrange- 
ments of lenses are most suitable for new optical 
instruments. He begins by saying that he wishes to 
make an enumeration of the improvements which art 
can supply, after enumerating the natural provisions : 
(i) bodies, the objects of vision; (ii) the interior organs 
receiving the action of these bodies, and (iii) the 
exterior organs, the eye and the media of vision, 
which dispose these objects so that their action can 
be properly received by the inner organs. ^ 

Descartes, then, considers these three heads one by 
one and makes the relevant distinctions; he discusses 
possible hindrances and aids, setting aside those which 

1 Ibid., pp. 395-6 and pp. 411-25. 

2 p. 390. 

8 A. & T., VI, pp. 147! 

are clearly impossible and estimating what obstables 
and deficiencies cannot be removed by human know- 

These enumerations are clearly a preparatory map- 
ping of the whole province of study, in some corners 
of which special investigation is to take place, and the 
solution of special problems to be found. The enumer- 
ation ensures that we see and consider whatever, in 
the province of the science as a whole, may be 
relevant to the special inquiry. In the wider domain 
there may be links essential to chains of reasoning 
required to solve the special problem (or problems). 
We must not disregard any link, or our subsequent 
conclusion may be invalidated. 

But it is not easy to see how, on Descartes's theory, 
it is possible to make a preliminary survey at all. 
In the conduct of such surveys he does not say what 
our power of knowing is. A logical procedure is 
implied for which he does not seem to have allowed 
in his account of the intellect, as exhausted by intuitus 
and deductio. For preparatory surveys of this sort are 
of necessity general and abstract, and Descartes says 
they must be 'sufficient' that is, to guard against the 
omission of anything which will be relevant to the 
problem subsequently investigated. 1 Further, they 
must be ordered that is, conducted upon some prin- 
ciple indeed any principle would do, provided it gave 
a comprehensive, convenient and time-saving arrange- 
ment. The enumerations are, therefore, always to 
some extent, and usually in the main, skeleton out- 
lines; in fact, a sufficient enumeration need be little 
more than a disjunctive limitation of the gaps in our 
1 Cp. A. & T. X, pp. 589-90. 

knowledge, while the connections between the data 
are known only in the barest outline. 1 The matters 
listed in the enumeration are included only as general 
groups; and within each subordinate group there is a 
plurality of singulars. But this internal detail is not 
specified and may be, as yet, unknown (either to 
anyone at all, or to us, the investigators at the mo- 
ment), or may be known to be irrelevant to our 
special investigation and disregarded. So a general 
designation 'is all that need represent the group in our 
enumeration it may be only an indeterminate nega- 
tive; 'anything not-A' and by this general designation 
we circumscribe an area left blank for our purposes, 
as a gap in knowledge, or a gap simply in our present 
knowledge, or merely an area seen to be irrelevant. 
It is a device by which we mark off an enemy fortress 
so that we can continue to advance, though we have 
not yet captured it. 

In extreme cases a device of this kind would not 
be very helpful and would not enable us to pursue a 
profitable line of advance. Under some, or many, of the 
heads of the enumeration the subject matter may be so 
abstract or general that we cannot commit ourselves to 
any but superficial judgements. Descartes says 2 that 
only with the help of enumeration can we pass a 
certain judgement on any subject at all, and only with 
its help shall we know something about all the ques- 
tions in our science. But, in many cases, this 'some- 
thing' we should know may be so little that it is not 
worth mentioning. 

1 Cp. p. 390. 11. 13-18: Enumeration is merely auxiliary to the 
proof that the soul is not dependent upon the body. 

2 Ibid., p. 388. 

Enumeratio is not, therefore, a method of proof at 
all. Descartes gives the name to two distinct devices: 
one for retaining in the memory the data from which 
inference has already been made; and the other a pre- 
liminary survey of the ground to select, compare and 
arrange the materials for an inference which is about 
to be made. The inference itself is always a movement 
of illation from one simple to another by the intellect 
deductio. This passage (or transition) constitutes a 
linkage (or implication) which is self-evident, as are 
the inter-linked simples themselves. The linkage, in 
fact, is in principle identical with the nexus between 
implicans and implication within each single self- 
evident proposition. 

Against this interpretation it may be said that Des- 
cartes frequently speaks of inductio, and, in one pas- 
sage, of imitatio, suggesting that these are modes of 
proof other than deductio or intuitus. But he expressly 
identifies the power of knowing with deductio and 
intuitus alone. 1 Also he identifies inductio with special- 
ly intricate form of deductio, where the premises are 
complex or confused; and he gives no explanation of 
imitatio in connexion with inductio. 

The one passage in which imitatio is mentioned 2 
is very difficult, and since it has a bearing upon 
Descartes's method, we have good reason to examine 

In actual inquiries we may sometimes come upon a 
link the implications of which, and therefore its im- 
mediate logical successor in the chain, are obscure, 
and advance is accordingly barred. There are two 

1 Reg. xii, p. 425, 11. 10-12. 

2 Cp. Reg. viii, pp. 393-5- 


possible varieties of such situations: (i) The barrier 
may be absolute and insuperable owing to the limita- 
tion of the human mind. 1 (ii) The barrier may be in- 
superable only for those who confine themselves with- 
in the limits of a special science, but may be sur- 
mounted by those who pursue the universal aim of 
science and follow the principle of the Cartesian 
method. As stated under Rule I, the student's interest 
ought not to be confined to a single science, but he 
ought to study all. Descartes gives as an example the 
following problem: 

Suppose a student whose interest is confined to pure 
mathematics sets out to discover the line of refraction 
in optics. Such a student will follow the method of 
analysis and synthesis, set out in Rules v and vi, and 
will see that the determination of this line depends 
upon the relation between the angles of incidence and 
thd angles of refraction. 2 He will recognize that the 
discovery of this requires a knowledge of physics and 
is impossible for the pure mathematician, so he will 
break off his inquiry. 

But now suppose a student whose aim is universal : 
he will desire to pass a true judgement here also, 
and, as a genuine student of all the sciences, he will be 
able to complete the analysis, proceeding until he 
reaches the simplest link in the implicatory sequence 
involved in the problem. He will find that the ratio 
varies in accordance with the variation in the angles 
resulting from changes in the physical media, and that 
these again are dependent upon the mode of propaga- 

1 Vide Reg. v'm, p. 396. No example is given though he says that 
many such cases may occur. 

2 Cp. La Dioptrique A. & T. VI, pp. 100-1, 211-214. 

tion of the rays of light. He will find that knowledge 
of this propagation requires a knowledge of the nature 
of illumination; and that again presupposes the know- 
ledge of what a natural force, or energy, is. In the 
last presupposition the student has reached the simp- 
lest link in the implicatory sequence (for this problem), 
and, having gained a clear insight, he will now (in 
accordance with Rule v) begin his synthesis link by 

If now he finds himself unable to perceive the 
nature of illumination, he will enumerate all the other 
forms of natural force (in accordance with Rule vii), 
so that he may understand, at least by the analogy 
'imitatio') of what he knows of the other natural 
forces. Descartes promises a later explanation of imi't- 
atio, but this promise remains unfulfilled, probably 
because the Regulae was never completed. He seems 
to have in mind a process of hypothetical construction, 
in this instance, of the nature of illumination by 
analogy from the nature of some other natural force 
which is known to the inquirer and which it may be 
supposed to resemble. 1 

1 Cf. Reg. xiv, A. & T. X, pp. 438-9. 



RULE iv: We have next to consider why a method 
is necessary for investigating the truth of things, what 
it can hope to do and on what rules it should proceed. 
The method is explained and justified in the first half of 
the exposition of Rule iv. 1 Owing to the origin of 
Descartes's conception of method he tends to confuse 
it with science and is led to speak of his new science of 
order and measure. 2 The second half of the exposition 3 
is devoted to an account of this, and here Descartes 
^xplains how he himself came to discover it, and the 
passage is largely autobiographical. In the Hanover 
MS. it comes at the end of the Regulae, but even there 
a note to Rule iv refers to it. 4 

Descartes tells us in the exposition of Rule vii that 
Rules v, vi and vii should be taken together; that all 
three contribute equally to the perfection of the meth- 
od, and that 'the rest of the treatise* (presumably 
Rules viii - xi, for from Rule xii onward a different 

1 A. & T., X, pp. 371-4. 

2 Cp. below pp. 8 iff. 

3 Ibid., X, pp. 374-379- 

4 Ibid., p. 374 n.a. 

subject matter is treated) does little more than work 
out in detail what they cover in general. In the dis- 
cussion of enumeratio we have already dealt with 
Rule vii, so we may now consider the exposition of 
Rule iv 1 and Rules v and vi. 2 

Descartes says that nearly all chemists, most geomet- 
ers and the majority of philosophers pursue their 
studies haphazard without any method at all. Occasion- 
ally these aimless studies lead accidentally to truth, 
but these coincidental cases are more than outweighed 
by the serious injury done by such procedure to the 
mind, because vague and confused studies weaken the 
natural light. That is why people with little or no 
learning are often far superior in forming clear and 
sound judgements on the ordinary matters of life. 

By a method he means certain and easy rules such 
that anyone who precisely obeys them will never 
take for truth anything that is false, and will advance 
step by step, in the correct order without waste of 
mental energy, to the knowledge of everything that 
he is capable of knowing. But he suggests that it is not 
the method which enables us to know, for it is the 
nature of the intellect to perceive what is clear and 
distinct and also to move infallibly from one self- 
evident link to the next, along the line of logical 
implication. Not only are intuitus and deductio the 
"Sole means of knowing this they must be in order to 
be vis cognoscens at all but we could not learn to 
use them, because we should have to use them in 
order to learn, and unless we both possess and use them 
from the first we could get nowhere. But, says Descar- 

1 pp. 371-2. 

2 pp. 379-87- 


tes, if we use the method, we can increase, guide and 
exercise the powers of intuitus and deductio. We can 
so arrange the materials, on which our power of 
intellectual vision is to be turned, and the links which 
are to form the stages of illation, that our natural 
powers will work under the most favourable condi- 
tions and will develop in scope and in intensity. 

The general principles of the method are set out in 
Rules v, vi and vii and the details in Rules ix, x and 
xi: Those applying to intuitus come under Rule ix, 
those concerning deductio, in more complex cases, 
under Rule x, and those applying to cases where both 
together and concomitantly are to be used, under Rule 

The entire method consists in the ordering and 
arrangement of material on which attention is to be 
Concentrated. Exact observance of it will be secured if 
we reduce involved and complex propositions to sim- 
ple ones and begin by exercising our intellect (in- 
tuitus) 1 on the simple, and then work our way up step 
by step to all the others. In the exposition of this 
Rule, 2 Descartes emphasises its supreme importance. 
Here as elsewhere, he warns us against the neglect of 
the simple and easy. Many scientists, he says, have a 
tendency to attack the most intricate problems before 
they have resolved the more elementary difficulties; 
and philosophers also neglect the obvious facts of 

But though this advice is undeniably sound, the 
difficulty is to carry it out. How can we reduce the 

1 'Ex omnium simplicissimarum intuitu' means 'insight into the 
simplest propositions of all', not 'all the simplest propositions'. 
Cp. Reg. ii, p. 364, Keg. in, p. 368 and Reg. x, p. 401-3. 

complex to the simple ? How are we to recognize the 
degrees of simplicity in the results of our analysis? 
How may we arrange them in the right order and 
know what is the simplest proposition of all when we 
reach it? Rule vi professes to give us the answer. 1 
At this point the doctrine of the Regulae has taken a 
new turn, and the details of this new development 
are difficult and obscure. Its general tendency is hard, 
if at all possible, to reconcile with the theory which 
has hitherto been attributed to Descartes and into 
which at last he unintentionally relapses. So far, Des- 
cartes has said that in every investigation our aim 
must be to transform all complex propositions 2 into 
an intelligible combination of simple elements, so 
clear to the intellect that they can be directly known. 
We must reconstitute the complex in terms of the 
simple and self-evident. Every proposition must be 
resolved by analysis into simple or elementary consti- 
tuents, and then resynthesized in an order which 
makes their relations and implications transparent to 
the intellect. This doctrine is a familiar one in philoso- 
phy, but the criticism of it is also familiar. The kind of 
analysis demanded is usually, or perhaps always, im- 
possible, not simply because of the weakness of our 
intelligence, but in the nature of the case. For it is not 
true that all, or most, complex facts could be known 
ff treated in this way. Few complex facts, if any, are 
sums or combinations of isolable and externally re- 

1 Vide pp. 381, 1. 8 and 382, 11. 17-19. 

N.B. Baillet omits Rule vi in his popular summary of the 
Regulae. In Adam & Tannery his words on Rule v are mis- 
takenly referred to Rule vi. 

2 Austin has 'every complex problem'. (Ed.). 

E 65 

lated constituents and to conceive them as such would 
not be knowledge but error. But how are we to recon- 
cile with such a theory the doctrine of method now 
to be maintained, a doctrine underlying Rule v and 
worked out in Rule vi? 

According to this doctrine the method still, no 
doubt, consists as before entirely in analysis and syn- 
thesis, but we now learn that these operations are 
gradual and their results graduated. It is not simples 
that are combined to form a complex. The complex is 
the last in a series of terms which gradually grow in 
complexity : it is the end of a long development at the 
beginning of which is the simple, or alternatively, the 
simple is the limit of a process of progressive reduction 
or gradual simplification of the complex, into increas- 
ingly simple propositions and terms, continued until 
the ('absolute') simplest proposition of all is reached. 
Synthesis starts from this and advances through the 
stages of the reductive (analytic) process in the re- 
verse order. So it issues in more and more complicated 
propositions and terms until it ends in the original 
complex but now reconstituted as a clearly intelligi- 
ble and demonstrated conclusion deduced from the 
early less complex propositions and terms. 

Thus, according to these new rules, the process of 
analysis is continuous and proceeds to the absolutely 
simple, while the synthesis is a progressive reconstruc- 
tion beginning from the simple and going through the 
increasing degrees of complexity until it ends in the 
original complex from which the analysis started. 

So far as we can judge, this new conception of 
degrees of simplicity or complexity is not consistent 
with the former conception which Descartes has been 

maintaining and which he, nevertheless, continues to 
maintain to the end. There is only one field where the 
two views seem, and perhaps are, compatible that of 
number, from which alone Descartes takes his exam- 
ples. 1 

RULE vi. The Latin here is faulty but the meaning is 
clear. If we are to distinguish the most simple things 
from the most complex and to advance in the right 
order, we must proceed as follows: taking any 
sequence of truths deduced one from another, we must 
observe which is the simplest and how the rest are 
related to it whether more, or less, or equally re- 
moved from it. 

The exposition is obscure. As the Rule states, 
things can be arranged in certain series. They are not 
to be considered as separate and independent in char- 
acter, nor is the arrangement of them in series to be 
regarded as a grouping under Aristotelian categories 
or different kinds of being. What is meant is the 
connection of things in their logical relations, so far as 
the knowledge of one can be derived from that of 
another (or others), things related qua implicantia 
and implicata. The series are implicatory sequences. 

In every such series the terms may be distinguished 
according as they are either absolute or relative, indep- 
endent or derivative. If relative or dependent they 
may be distinguished in respect to the kind and degree 
of their dependence. In each sequence there is a first 
term, on which all the rest depend but which is 
independent of them; a second term dependent on the 
first but not on the rest, though all others depend upon 
1 Vide pp. 384-7 and 409-410. 


it. The implication is always unilateral and not recip- 
rocal. A proposition (for Descartes) may be necessary, 
even though its converse is contingent. 1 

Further, 'absolute' and 'relative' turn out to be 
themselves relative terms in this connection. A term 
may be absolute from one point of view and relative 
from another. It may be more or less absolute or 
relative. 'Absolute' and 'relative* are to be understood 
with reference to the subject under investigation. 

A term is absolute in the highest sense, if it contains 
the 'nature' 2 under investigation in its purest and most 
simple form. This is the limit of the analysis and so the 
first term in the re-synthesis. So in the series of steps 
leading to the discovery of the line of refraction, the 
last term of regressive analysis or first of the progres- 
sive re-synthesis is 'natural energy'. Now this is 
absolute in relation to the second term, illuminatio, 
because it stands to it as universal to particular. 
'Natural energy' or 'force in general' is an abstract 
universal in which heat, light, etc., etc., all share in 
various ways. But in other examples, the contrast 
between two terms as absolute and relative may de- 
pend on one or other of a variety of antitheses 3 : 
simple as opposed to complex, similar as opposed to 

1 Cp. Reg. xi, p. 422. The relations of successive terms in such an 
implicatory sequence are indicated by Descartes and Spinoza 
by certain technical terms. The absolute and independent & 
related to the relative and derivative as 'Substance' to 'Mode', 
and the relatively more independent to the relatively more de- 
pendent as 'primary mode' to 'subordinate mode'. 

2 Descartes's use of the term 'natura', here and elsewhere, is vague 
(vide pp. 382, 11. 3-4; 383, 1. 3; 440, 11. 10-20). It is perhaps 
reminiscent of Bacon. 

8 Cp. pp. 381-2. 

dissimilar, straight as opposed to oblique. But (and 
this is the real crux) whatever the antithesis, we must 
remember that the point of the contrast is always the 
order of logical implication. It is the absolutely or 
relatively known, not the absolutely or relatively 
existent, with which we are concerned. Thus in the 
case of cause and effect, in existence they are corre- 
lative, but in knowledge the effect presupposes the 
cause, which is thus prior in the implicatory sequence. 
Similarly unequal, dissimilar, oblique, presuppose the 
knowledge of equal, similar, straight (though not their 
existence or reality) and so come before them in the 
implicatory series; i.e. they are r& \oyu var^u, KW 
ttpoTtpa, (Descartes clearly has Aristotle's distinction 
in mind). 

In every investigation of a complex problem, there- 
fore, we must gradually reduce it to the most absolute 
term, pressing on with the regressive analysis until we 
reach the term which is first in this sense. On this we 
must concentrate our mental vision until we have a 
perfect and thorough mental insight. We can then 
begin a reconstructive synthesis, correctly arranging 
the terms of the implicatory sequence, which issues in 
the solution of the problem. 

All the terms, except the first, are called by Descar- 
tes 'relative' ('respective'), because every one of them 
implies the first, and the conception of them presup- 
poses as a condition the 'real' or 'nature', which is em- 
bodied in the first or absolute term. They are derived 
from it by a continuous deductive chain and embody 
it in part. But they include also other elements or 
features, and if we are to conceive them adequately, 
we must take these added determinants into account 


as well as the absolute nature. Descartes calls these 
other elements 'respectus' and they are dependent 
upon relations to other things in other implicatory 
sequences. Thus even the least relative of the relative 
terms contains, in the adequate conception of it, a 
feature that restricts the simple nature of the absolute 
term. This feature is a respect, or regard, connecting 
it with (and making it relevant to) another sequence. 

As the synthetic series proceeds, the successive terms 
grow in complexity and each contains more and 
more of these respectus, or references beyond itself; 
they grow in complexity or concreteness. But this 
means that there is a continuous addition of fresh 
determinants more and more features are added as 
the series advances and each of these is a relative 
term connecting it with an absolute belonging to an- 
other sequence. So every relative or complex term in 
a series is the meeting-point of two or more implica- 
tory sequences, each of which, if followed up, would 
lead to its own absolute term, 1 and the mind must 
combine a number of features in conceiving each of 
the relative terms. 

So the analysis and synthesis of Descartes's method 
yield a number of implicatory sequences, each invol- 
ving a number of terms which grow in complexity as 
the sequence advances. In each the first term is a 
simple nature a certain feature of reality in its 
purity and each successive term is derivative from 
this first one. The subsequent terms each contain this 
along with other added features which relate the first 
to other, different implicatory sequences. In other 

1 Vide, p. 382, 11. 3-1 6. 

words, there are (i) the simplest, primary and most 
absolute terms, and (ii) a number of increasingly less 
simple, more complex, derivative terms. 

The first point to be criticized is that the contrast 
between absolute and relative terms in every sequence 
is, in fact, based on one principle only (though Descar- 
tes says that there are more). It is always by subtrac- 
tion that the terms become more simple in analysis 
and can be arranged in a progressively more simple 
series, and it is only by addition that they become 
progressively more complex in synthesis. If X is a 
simpler term and Y more complex, Y will always 
contain X plus something else in addition. If X's 
character is precisely a, then the character of Y is a 
-f 3; and if there is a third term Z, more relative and 
complex than Y, this can only mean, according to 
Descartes, that the character of Z is (a + j8) + y. 

Descartes enumerates, it is true, several apparently 
different antitheses and asserts that any one may serve 
as the basis of the distinction between absolute and 
relative terms in a given sequence. It may be that Y 
is related to X as particular to universal, or as effect to 
cause, or as composite to simple, or as oblique to 
straight. But, in fact, his account of the order of terms 
always assumes that they are related to one another 
according to the above scheme: i.e., a; a 4- (3; 
(a-h j3) + y., etc., as if the only antithesis that 
could serve as a basis for relativity were between the 
abstract universal and that same abstract universal 
plus added determinants. 

This result seems to follow inevitably from Descar- 
tes's statements and it suggests three comments: 

(i) In the synthetic implicatory sequence the succes- 


sive terms do not become less simple, or more complex 
as the sequence proceeds, in any genuine sense. They 
may appear so, but when examined they will always 
reduce to absolute simples related by an absolutely 
simple connexion. It is always a case of less or more 
constituents and the more numerous constituents do 
not make the relative term really more complex as to 
the nature of the constituents or the manner of aggre- 
gation. The larger aggregate is not more developed in 
the mode of relation of its elements. Successive terms 
do not grow in concreteness, they are not genuinely 
one, or whole at all. Further, they do not even exhibit 
increasing complexity of structure. For example, 7 is 
not more complex than 6, if 6 is nothing more than 
the five-fold addition of i to i and 7 the six-fold 

So we come back to the original form of Descartes's 
method. It remains, as always, the resolution of a 
complex into absolute simples simply related a sum 
or aggregate. If any term in the process of synthesis or 
analysis seems to be more simple than another, that 
only means that the work of the scientist or philo- 
sopher has not yet been completed. We are grasping 
confusedly what, if perceived clearly, would be seen as 
an aggregate of simples, and confusing it with a gen- 
uinely concrete fact that is, a term genuinely single 
(in the sense that it can't be split up without being 
altered) yet not atomic. 

(ii) At every step there is a break in logical contin- 
uity. This is obvious in the synthetic reconstruction of 
the sequence, for each successive term adds a feature, 
to its predecessor, new in the sense that it is logically 
discontinuous and irrelevant to what went before. 


And this new 'respectus' refers the mind away along a 
different implicatory sequence to a different absolute 
term which the new determinant involves. 

(iii) To what field or fields of fact (or reality) can 
such a method be applied with any prospect of yield- 
ing knowledge? What kind of facts could it explain 
and not explain away ? Obviously, it can apply only to 
a field where every fact is either atomic and simple or 
an aggregate of simples. Such a method, then, is 
powerless to deal with such matters as the phenomena 
of life, and a fortiori with the domain of philosophy 
the field of conscious, or self-conscious, spirit or 

At first sight the objects of arithmetic and algebra, 
number and numerical proportion only, seem to sat- 
isfy the required conditions. They alone are such as 
to be explained, and not distorted or explained away, 
by the sort of reasoning guided by the rules of the 
Cartesian method. All applied mathematics, geometry 
and the whole of the rest of nature, animate and 
inanimate, because they involve continuity and move- 
ment, must elude the grasp of the method, and what 
Descartes says is sound reasoning. As for the spiritual 
facts of ethics, aesthetics or logic and all that are 
studied by philosophy, it seems ludicrous to suppose 
that any light could be thrown on them by a reason- 
ing based on the assumption that they are aggregates 
of simples. If so, the method is worthless in philoso- 
phy and of little or no use in any science except 
arithmetic and algebra. 

But we must consider what can be said against this 
criticism and we shall find that a strong case can be 
made both on general grounds and by reference to 


Descartes's own view of the relation between mathe- 
matics and philosophy 

1. General objections to the criticism. Many philoso- 
phers besides Descartes and all, or most, men of 
science would agree that analysis can be nothing else 
than the resolution of the complex into simples. 
Doubtless there are facts in our experience which 
stand out against such treatment and resist analysis; 
which if so treated seem to have been explained away 
But that means (they say) only that some facts are 
inexplicable and must always remain unintelligible 
just because they cannot be analysed. Or they might 
say that it may mean, or does mean in most cases, that 
some facts are so complex that they are beyond our 
present powers of analysis outside the reach of the 
longest chains of implicatory sequence we can con- 
struct. Nevertheless, they would claim, no more is 
needed than patience and persistence for analysis along 
the same lines to resolve even such complex cases. 
For, they would ask, what kind of analysis other then 
this could there be? What new method of procedure 
can be conceived? If there is such a method, it must be 
one which explains the concrete fact without analysis 
or resolution into simples, and it should be produced. 
In effect, the critic is challenged to show that any 
mode of reasoning, other than analysis by subtractioa 
and synthesis by addition, is possible. 

2. Descartes's special rejoinder. Descartes not only 
shares the position on which these general objections 
are founded, but he also maintains a theory of mathe- 
matics and of its relation to philosophy which cuts the 


ground from under the feet of the critic and seriously 
undermines his position. According to this theory, 
the method controlling mathematical reasoning, in its 
purest, most abstract and general form the proper 
method of the science as Descartes conceives it is eo 
ipso the method which our reason must use for know- 
ledge in every field of the knowable. In the case of one 
field only does Descartes make an exception that of 
self-conscious mind. Notions of extension, motion, 
figure, number, etc., comprising the whole of nature 
up to and including man's animal nature fall within 
the scope of mathematical reasoning. The sciences and 
the branches of philosophy which are concerned with 
these fields of being are nothing but off -shoots and 
developments of the supreme science; they are the 
expansion of the same spirit which produces as its first 
fruits the universal principles of order and measure, 
which Descartes identifies with true mathematics, and 
which he calls 'mathesis universalis' , as opposed to 
' mathematica vulgaris. 

The student of philosophy who is not a mathemat- 
ician hesitates to discuss this subject, but it is one of 
fundamental importance for the understanding of Des- 
cartes's philosophy, especially the Regulae; and any- 
body who wishes to make a serious effort and to form 
a just estimate of his doctrine must deal with it. 

What were the aim, scope and value of this univer- 
sal science ? Without the answer to this question even 
the exposition in the Discourse on Method 1 is not 
clear, and a fortiori that in the Regulae is not. Also, 
the primary significance of Descartes's theory is philo- 
sophical, and the strict mathematician, in his admira- 

1 Cp. A. & T., VI, pp. 17-20. 


tion of the technical details, may fail to see the wood 
for the trees. 1 

Descartes 's conception of method was derived orig- 
inally from reflection upon the procedure of arithmet- 
icians and geometricians. He was deeply impressed by 
the self-evident certainty of their reasoning and the 
ease with which it could solve abstruse, complex and 
intricate problems. He thought he had discovered the 
principles to which it owed its success, and his for- 
mulation of what he took these principles to be con- 
stituted his method in its earliest form. But he soon 
realized that his own methods, which he had used in 
doing this were much better than the actual reasonings 
of the geometers and arithmeticians, by reflection upon 
which he had formulated the principles. In these 
reasonings he became aware of defects; they were 
not logically continuous movements of thought along 
single lines of implicatory sequences, and the existing 
mathematical systems were not logically consistent 
and coherent as he conceived science should be. 2 

The substance of his criticism is that the proofs are 
of the 'mousetrap' variety the reader is tricked into 
agreement by some careless admission or even some 
extraneous absurdity imported into the argument,and 
is not really convinced. The argument does not incl- 
clude, as its middle term, the real bond of connection 
between premises and conclusion the real nexus 
inherent in the subject-matter its appeal is to the eye 
or the imagination rather than to the intellect and if 

1 Cp. Liard, Descartes (Paris, 1882), pp. 8-10, 35-63. 

2 Vide supra pp. 25-28, and Cp. Descartes's dissatisfaction with 
Mathematica Vulgaris, both as to subject-matter and method, 
expressed in Reg. iv., Expi pp. 374-8. 


the student is convinced, it is not by the so-called 
proof, but by his own independent insight. Hence 
education based on arithmetic and geometry alone, by 
concentrating attention on such spurious and super- 
ficial demonstrations, 1 actually tends to weaken the 
intellect by allowing its natural powers of intuitus 
and deductio to atrophy. 2 Descartes is also severe in 
his condemnation of the futility of the problems actu- 
ally set in the contemporary mathematics. 8 Nothing, 
he says, can be more futile than to devote oneself to 
the study of bare numbers and imaginary* figures, 
as though the whole aim of life were the knowledge of 
such things. 5 His point is that such studies are only 
valuable as a preliminary to physics or the philoso- 
phy of nature the more concrete study of the natural 
world, though mathematicians make extravagant 
claims for their collections of increasingly intricate 
problems and ingenious solutions, which are of no 
more than technical interest, and make no contribu- 
tion to furthering the knowledge of nature. 

But the defects of the contemporary mathematics, 
Descartes is convinced, were, so to say, accidental. 
The fault lay not with mathematics but with the 
mathematicians not in the essential nature of the 
science, but in the mistakes of individual scientists. 
He believes that there is a vera mathesis which, being 

l "Vide Reg. iv, p. 375. 

2 Cp. Schopenhauer's 'Four-fold proof of sufficient reason'. But 
his position is different in so far as he holds that, in geometry, 
the only adequate and genuine proof is, and must be, the appeal 
to spatial intuition. 

3 Reg. iv, pp. 371 and 375. 

4 Meaning 'imaginable' or 'pictured in the imagination'. Ed. 

5 Vide p. 375. 


the simplest, is the foundation of all knowledge a 
mathematics which is the necessary propaedeutic for 
the mastery of science and philosophy by the mind. 
That there must be such a science existing, so to speak, 
in posse, waiting to be discovered, follows, Descartes 
thinks, from the very nature of his method, and its 
existence, in some manner, must have been known, 
at least in outline, to the great mathematicians of 

For this method, two conditions must be fulfilled : 
we must accept as true only what is self-evident, and 
we must follow the correct logical expansion of these 
data. Only thus will the intellect be following the 
procedure dictated by its own true nature. This is 
consistent with another conviction of Descartes's, 
that there are certain seeds of truth implanted by 
nature in our minds; 1 and he speaks of 'inborn princi- 
ples of method' bearing 'spontaneous fruits. 2 They are 
seeds which tend to ripen spontaneously in a natural 
harvest of knowledge, that tends to develop along 
certain lines, which, if made definite and formulated 
as rules of guidance, are the Cartesian method. We 
must note, in passing, the following important points : 

(i) In the exposition of Rule iv, the development 
of knowledge is expressed in terms of the mind's atten- 
tive observation of ideas in itself and their natural 

1 Cp. Reg. iv, pp. 373 and 376, and Discours (A. & T. VI), p. 64. 
The doctrine may be traced back to Aquinas, De Veritate : 
'Praeexistant in nobis quaedam scientiarum seniina' (Quaes- 
tiones Disputatae, De Veritate, Quaestio xii, art 1.). 

2 Loc. cit. (p. 373) : '. . . spontaneae fruges ex ingenitis hujus 
methodi principiis natae . . .' 


expansion and development. The 'seeds' are implanted 
and are data given to the immediate seeing of the 
mind. As they are given they are also received, and 
reception is a determinate mode of the mind's innate 
functioning to see this, is to see it thus (somehow). 
Thus Descartes has already formed and is already 
working with the same conception of 'idea' as ap- 
pears in his later teaching. 1 

(ii) The emphasis laid here on growth and the meta- 
phor of seeds ripening to maturity take the place of 
the metaphor of links in a chain. We must not, how- 
ever, rashly assume that Descartes would have regar- 
ded increment or growth as anything other than addi- 
tion. Aristotle distinguished uv&crig from TxpoadtGig 
but we have no reason to suppose that Descartes 
would have done the same. 

He asserts in the exposition of Rule iv, 2 that so far 
as the easiest of all sciences (Arithmetic and Geometry) 
are concerned there is positive evidence that germin- 
al truths in the mind have spontaneously ripened 
and produced a harvest of knowledge. The old Greek 
geometers employed analysis in all problems, though 
they jealously kept the method secret; and the modern, 
flourishing algebra also attempts to apply to numbers 
the same sort of analysis as the Ancients applied to 
figures. Later, 3 he says more positively that traces of 
the vera mathesis are apparent in the works of Pappus 

1 Cp. esp. the end of answer to the Second set of Objections, 
A. & T. VII, p. 1 60. 

2 A. & T., X, p. 373- 

3 Ibid., p. 376. 


and Diophantus. 1 But the analysis of the old Greek 
geometers and the algebra of Descartes's time are 
very imperfect anticipations of his own vera mathesis, 
as he himself observes in the Discourse on Method. 2 
He complains that the ancients are so tied down to 
special figures that their work too narrowly restricts 
the exercise of the mind; and that the contemporary 
algebra seems, with its obscure symbols, rather to 
embarrass the mind than to clarify. This passage in the 
Discours is obviously based on Rule iv, 3 stating the 
same opinion more clearly, in a shorter, more elegant 
and more popular form. 4 In the passage of the Re- 
gulae? he says that algebra lacks the supreme clear- 
ness and facility which should characterise vera mat- 
hesis as a genuine embodiment of the method. 

Vera Mathesis. What, however, is this vera ma- 
ihesisl As Descartes describes it and as he elaborates 
and uses it for his philosophy of nature, it is an 
amalgamation of two heterogeneous elements, which 
are contributed by two different and discrepant facul- 

1 Cp. Heath, History of Greek Mathematics II, pp. 400-401, for 
Pappus' definition of analysis. Hamelin says (Le Systeme de 
Descartes, p. 7) that Vieta was responsible, more than any other, 
for casting geometrical problems in the form of equations; but 
Descartes did not read him until 1629 (after the Regulae had 
been written). Vieta owed most to Diophantus. 

2 Cp. A. & T., VI, pp. 17-18 and p. 549. 

3 A. & T., X. pp. 375-7- 

4 The Latin version of the Discours is better than the French. 
Descartes himself corrected and revised it and requested that 
it be regarded as the original. 

5 Reg. iv, p. 377. 

(i) As a science proper it is purely intellectual (as, 
for Descartes, all science must be) intellect express- 
ing and fulfilling itself in act with a purely abstract, 
intelligible domain. But (ii) the imagination co-operates 
with the intellect and, though Descartes regards it as 
only instrumental to the work of the intellect, yet to 
the imagination and its contributions are due the 
whole value of the vera mathesis for the science of 
nature, and its entire originality. 

Descartes's conception of vera mathesis, considered 
as a science, and of its domain as the subject-matter 
of a science, is stated very clearly in the autobiograph- 
ical passage in the exposition to Rule iv. 1 He says 
that when his thoughts turned from the special scien- 
ces of arithmetic and geometry to the idea of univer- 
sal mathematics, he first asked himself what is meant 
by the term 'mathematics', and why many sciences, 
such as optics, mechanics, astronomy, etc. are com- 
monly reckoned as parts of mathematics. What is 
common to all these despite their different subject- 
matters? How can the beginner at once see what 
belongs to mathematics what makes an investiga- 
tion mathematical? After careful consideration, he 
says, he came to the conclusion that inquiries in which 
order and measure are examined, and these alone, are 
referred to mathematical science it matters not 
whether in numbers, figures, sounds or stars. There 
must, then, be a science relating to order and measure, 
as such, in general, and abstracted from their relation 
to this or that special subject-matter; and this alone 
is entitled to be called mathesis universalis or vera 
mathesis. Geometry, arithmetic, etc. are only called 

1 PP. 377-8. Cp. Reg. vi, p. 385* 11- i-4- 

F 81 

mathematics in so far as they each deal with a part of 
the domain of universal mathematics. 

The same doctrine is expressed more shortly in the 
Discours. 1 All the mathematical sciences, he says 
there, are concerned with a common object of inves- 
tigation : the relations or proportions obtaining with- 
in their special subjects. The domain, then, of pure 
mathematics is that of proportion as such or in general. 
Vera mathesis or mathesis universalis is, therefore, the 
Cartesian method directly applied to a systematic 
investigation of all problems connected with propor- 
tion, order and measure, conceived as such and in 
general, in abstraction from the particular things in 
the subject-matter which bear these proportions. 

The terms used in both the Regulae and the Dis- 
cours 'order', 'measure' and 'dimension', are technical 
terms in the new vera mathesis. Descartes gives a 
somewhat sketchy account of them in the exposition 
of Rule xiv. 2 'Dimension' is any aspect of a perceptible, 
picturable or imaginable object in respect of which it 
is measurable; including, for example, weight and 
velocity (the dimensions of motion). Order applies to 
a manifold (Mengef and measure only to a continu- 
ous magnitude. The latter can always be reduced to a 
manifold, saltern ex parte, by the help of an assumed 
unit, and the many so obtained can then be ordered 

1 A. & T., VI, pp. 19-20, 550-1. 

2 PP. 447-52. 

3 Austin's version has 'manner', which is surely due to a mis- 
hearing of the German word 'Menge'; though why Joachim 
should have drawn attention to the German equivalent is not 
clear. Presumably, he had Kant's usage in mind : cp. Kritik der 
Reinen Vernunit, A.ios, 163 and 8.204. (Ed.). 


in such a way as to facilitate measurement of the 
magnitude. In the more popular summary in the 
DJ scours, Descartes substitutes the less technical term 

So far, there is nothing original in Descartes's vera 
mathesis and nothing very promising for the solution 
of problems in the physical sciences or for their future 
development. The idea of mathematics so specialized 
is familiar to Aristotle, 1 who makes it clear that the 
Greek mathematicians of his day had developed a 
theory of proportion in general. He refers to the 
theory that proportions alternate and says that this 
theory used to be demonstrated in detachments for the 
different species of proportionate things (numbers, 
lengths, durations, etc.) But (says Aristotle) alternation 
is true of all proportionals, in virtue of their common 
character and is not dependent on the features in 
which they are specifically distinct from one another. 
Hence, nowadays, (he continues) the mathematicians 
postulate something present in all proportionate things 
and say that alternation is characteristic of this some- 
thing, being universally predicable of it. 

Reverting, then, to Descartes's attack on the vul- 
gar mathematics, we can now see one feature of his 
criticism more clearly. The intelligible domain of 
mathematics the only proper subject of the vera 
mathesis, as a science is proportions qua propor- 
tions, or proportion in general, which does not alter 
with the subjects which it informs, and must be ab- 
stracted from its particular embodiments. Among these 
subjects are numbers and figures; but the vulgar mathe- 
maticians, says Descartes, miss the substance and pur- 
1 Metaphysics 1026326-27, and An. Post. 74317-25. 


sue the shadow. 1 On the other hand, the Greeks and 
the contemporary algebraists did attempt to grapple 
with the proper subject of the science and with what 
is embodied in figures and numbers, but they failed to 
realise that proportions, as such, have nothing to do 
with the figures and numbers in which they happen to 
be enwrapped. There is no need to be tied down to 
figures (like the Greeks) or to numbers (like the alge- 
braists). So in place of a general theory, these thinkers 
produced theories appropriate to each separate field, 
and they were able really to grapple with proportions 
only in those two domains of number and figure. The 
proper business of vera mathesis, however, is to treat 
proportion in abstraction from numbers and figures as 
well from all other embodiments. 2 

The role of the imagination in vera mathesis is 
summarised in the Discours, 3 where, though Descartes 
skates over the difficulties, he gives a clearer general 
view than in his more elaborate and contorted account 
in the Regulae. He says that he is determined to 
study proportions in general without referring them to 
any objects in particular; but he intends to use some 
objects to aid the intellect and facilitate understan- 
ding. He would, he decided, sometimes have to study 
each kind of embodiment of proportion apart from 
the others, and sometimes many at once, keeping them 
in mind by means of memory. When he comes to 
consider proportions separately, in abstracto, he thinks 
it best to study them 'tan turn in lineis rectis', for, 

1 Cp. Reg. iv, pp. 373 && fin- esp. pp. 374-7. 

2 Cp. Reg. xiv, p. 452, 11. 14-26, and Reg. xvi, p. 455 ad fin., (esp. 
p. 456). 

8 A. & T., VI, p. 20 and p. 551. 

he says, he can find nothing simpler than straight lines, 
or better adapted to represent proportions distinctly to 
the senses or the imagination. But to study many kinds 
of proportion together he had decided to use various 
symbols letters and numerals algebraical formulae, 
the notation of which he has explained elsewhere. 1 
This would embody all that is best in the Greek geo- 
meters, as well as the work of the algebraists, while it 
supplied the deficiencies and corrected the errors of 

But for a proper appreciation of the use of imagin- 
ation in vera mothesis we must refer to the more 
complicated account in the Regulae. 2 Descartes is here 
explaining the plan of the whole of the Regulae. He 
says that all possible objects of knowledge may be 
divided into simple proportions, on the one hand, and 
problems (quaestiones) on the other. The former must 
present themselves spontaneously to the mind. They 
are identical with data. There are no rules for the 
discovery of simple proportions, nor are they to be 
found by deliberate search. All we can do is to give 
the intellect certain precepts for training the powers of 
knowing in general, rules which will make it see more 
clearly and scrutinize more carefully the objects pre- 
sented to it. These are covered by Rules i-xii, while 
Quaestiones' are to form the subject matter of the 
remaining rules. Rules xiii and xxiv will deal with 
'quaestiones quae perfecte intelliguntur' (i.e. those 
fully understood, both as to their terms and their 
solutions, even though the actual solutions are not 
yet known) and Rules xxv onward are to deal with 

1 Cp. Reg. xvi, pp. 454-9- 

2 Reg. xii, pp. 428-9. 


'quaestiones quae imperfecte intelliguntur* (i.e. those 
problems not perfectly understood, but obscure in 
relation to some of the terms in which they are 
formulated and some of the conditions relevant to 

Descartes gradually unfolds his doctrine, starting 
from general considerations applying to both kinds of 
problem. 1 In every problem there must be something 
unknown, which must be somehow designated that 
is, referred to something known. This is true of all 
problems, perfect or imperfect. Suppose that we set 
out to inquire what is the nature of a magnet (imper- 
fect). The meaning of the terms 'nature' and 'magnet' 
must be known, and by these cognita the search is 
restricted and determined to a solution of a certain 

t All imperfect problems can be reduced to perfect, 
but the rules for doing this, which Descartes says will 
be given in the proper place, 2 were never completed. 
(We may conjecture how this may be done from 
what he says in the exposition of Rule xiii). We may 
take completed experiments and argue from them as 
fixed facts, or data, the imperfect question thus be- 
coming perfect. For instance, we may reformulate 
our previous question and ask what must be inferred 
about the nature of a magnet from Gilbert's expen- 
ments. We know the precise nature of Gilbert's ex- 
periments, and the quaesitum is now that solution 
which is individually determined by reference to these 
data and to them alone. What (we must ask) is the 
necessary inference (neither more nor less) from these 

1 Vide Reg. xiii, pp. 430-38. 

2 Ibid, p. 431 (i.e. Reg. xxv-xxxvi). 

cognita. But Descartes further maintains that not only 
can an imperfect problem be reduced to a perfect one, 
but every perfect problem can or ought to be further 
reduced until it becomes one belonging to the domain 
of vera mathesis i.e. one concerning proportions, 
order and measure alone. The reduction of a problem 
is not complete, or a problem is not strictly perfect, 
until it has become purely and abstractly mathemat- 
ical. So we must go on refining our cognita until we 
are no longer studying this or that matter in which 
proportions are embodied, but are concerned only 
with comparing sheer magnitudes. 1 Accordingly, Des- 
cartes says, pure or perfect problems occur only in 
arithmetic and geometry. 

The necessity of this reduction of problems, to 
questions of pure, universal mathematics, follows 
from the inherent nature of the intellect, 2 its limit- 
ations as well as its positive capacities. The exposition 
of Rule xiv, throws further light upon the distinction 
between absolute and relative terms in the implicatory 
sequence. Here he points out that obviously we cannot 
by sheer reasoning discover a new kind of being or 
nature. If from the known we deduce an unknown, 
all that our new knowledge involves is the perception 
that the unknown (res quaesita) participates, in this 
or that way, in the nature of the known data. To 
season to a new kind of entity would be as impossible 
as to argue a man blind from birth into perceiving 
true ideas of colours; though Descartes admits that it 
might be possible for a man who had seen the primary 

1 Vide Reg. xiii, p. 431, 11 15-27 and cp. Reg. xiv, p. 441, 11. 21-29, 
and Reg. xvii, p. 459, 11. 10-15. 

2 Cp. Reg. xiv, Expl. pp. 438-40. 


colours to construct for himself intermediate colours 
by 'a kind of deduction based on similarity' (imitatio?) 
If a magnet had a nature quite unlike any we had ever 
perceived we could reach it only by means of some 
new sense, or of a mind like God's. The utmost that 
the human mind could achieve would be to perceive 
distinctly that combination of known natures which 
would produce observed effects. 

According to the doctrine here implied, the intell- 
ect presupposes, as the condition of its deductive 
movement, an already existing knowledge of certain 
kinds of objects (or natures). Descartes has said that 
the mind must deduce from an immediately apprehen- 
ded, purely intelligible datum; but what he has in mind 
here, is not this, nor such a purely intelligible insight 
as he mentions in the Discourse on Method. The sort 
of knowledge he is thinking of, here, is a kind of 
sensuous or imaginative apprehension. In order to 
deduce, the intellect must start from and move within 
a nature that is known in the same sort of way as an 
object which is presented to sense. That this is his 
meaning becomes clear when one observes how the 
argument proceeds. In a case like that of the magnet, 
such previously known natures are, for example, ex- 
tension, figure, motion, etc., each of which is such 
that it is recognized in every object and is seen as 
the same idea in all its embodiments we picture 
(imaginamus) 1 the shape of a crown always by means 
of the same idea, whether it be made of silver or gold. 
This idea is transferred from one body to another by 
virtue of simple comparison, and the comparison must 
be 'simple and open', if the inferred conclusion is to 
1 Vide ibid., p. 439. 

be true. But the comparison of two things is simple 
and manifest only when they contain a nature equally. 
Hence, all inferential knowledge (i.e. all proper know- 
ledge other than intuitus) is obtained by the com- 
parison of two or more things; and if our knowledge 
is to be precise, we must so formulate and purify our 
problem that we perceive the quaesitum as like, equal 
to, or identical with, the datum in respect of some 
nature contained in both. 

Before the problem is properly formulated the terms 
are not directly comparable. Quaesitum and cognitum 
do not exhibit a common character which is seen to be 
equal by simple inspection and comparison. As the 
problem is first stated, the common nature is contained 
in the terms unequally or enwrapped in certain other 
relations or proportions. 1 Our main task, then, to 
which* we must devote ourselves if our reasoning is 
to give precise knowledge, is so to reduce these pro- 
portions that there may emerge to our view equality 
between the quaesitum and something else already 
known to us. A perfect problem presents only one 
kind of difficulty : namely, that of so developing the 
proportions that they may be disentangled from the 
qualities in which they are enwrapped. 

Now what exactly is it which, in this preparatory 
formulation of the problem, is being reduced to equal- 
ity? We can only answer: That which is susceptible 
of more or less. That is magnitudes magnitudes, in 
general and as such simply qua exhibiting degrees, 
or equality, and commensurable as the two sides of an 
equation. This is the object of our science in so far as it 
1 Cp. Reg. xiv, p. 440, 11. 15-16. 

is vera mathesis : so far as it is a rational activity of the 
pure intellect. 

Up to this point it seems as if Descartes's criticism 
of the problems of the vulgar arithmetic and geometry 
would apply a fortiori to those of vera mathesis. A 
science whose sole object is the comparison of magni- 
tudes in general, so as to make them equal or commen- 
surate, certainly seems to be engaged in the emptiest of 
tasks. What could be more futile than to equate 
amounts of nothing in particular! But in Descartes's 
own account of the matter this extreme abstractness 
and sterility of the domain of the vera mathesis is 
corrected or at least concealed by the part he as- 
signs to the imagination. He assumes that we have an 
imaginative knowledge of certain natures which is an 
indispensable condition without which the intellect 
cannot deduce at all. He takes for granted the sensuous 
or imaginative knowledge of extension, figure and 
motion the fundamental characters of the physical 
world as we perceive it. The magnitudes in general 
which the intellect studies and equates are abstracted 
from these. They are not, therefore, amounts of noth- 
ing, for the imaginative knowledge of one or more of 
many and various somethings must accompany every 
piece of scientific thinking, though we are to pay atten- 
tion only to their magnitudes. Descartes assumes, 
rightly or wrongly, but without question or discussion, 
that we can thus study or know magnitudes in general 
that the differences between what they are amounts 
of does not at all affect the amounts. He calls in the 
imagination merely as an aid to the intellect in its 
strictly scientific study of magnitudes as such. 

When we have reduced the problem to its mpst 


perfect, abstract, mathematical formulation, we must 
then transfer this to real objects in extension 1 and 
present it to the imagination as embodied in figures 
so that it will be perceived by the intellect with far 
greater distinctness. 2 Having first extracted the abs- 
tract mathematical substance, we are then to re- 
embody it in one special matter we are to make it 
once more an object of one special sort of imaginative 

This seems quite amazing, and seems to contradict 
the account given earlier of the intellect as related to 
sense and imagination. Is not pure intellect more pre- 
cise than intellect working through the organs of sense 
and phantasia? One would have thought that to en- 
wrap the purely intelligible object of science in any 
concrete embodiment must ipso facto diminish the 
distinctness of intellectual apprehension. 

No doubt Descartes means us to replace the concrete 
wrapping by one of a special kind which is the object 
of a special sort of imaginative knowledge. 3 For 
nothing can be said of magnitudes in general which 
cannot also be applied to species of magnitudes in 
particular; and there is one species of embodiment 
most easily reproduced and depicted in imagination. 
That is the real extension of body abstracted from 
everything except its shape. This can obviously be 
most easily and exactly represented by the arrange- 
ment of bodily parts of the bodily organ, phantasia. 
Other species differences of pitch in sound, or satura- 

1 Reg. xiv, p. 438. 

2 loc. cit. : 'ita enim longe distinctius ab intellectu percipietur'. 

3 Cp. p. 441. 


tion of colour cannot be so easily or precisely re- 
produced. 1 

Nevertheless the general effect of the doctrine of 
the Regulae concerning vera mathesis is to attach an 
overwhelming importance to imagination and its pic- 
turable objects. The object of science is to reduce pro- 
blems to equations of pure magnitude, but the intellect 
can do this only by making a preliminary abstraction 
from picturable, concretely embodied, magnitudes. 
It has to abstract amounts from natures presented to 
sense, in the first place; and secondly, when we have 
thus formed a conception of these magnitudes in 
general, we can do nothing with them unless we re- 
embody them in sensuous figures or at least those 
which, for this purpose, are simplest : namely, straight 
lines, rectilinear and rectangular figures. 2 Descartes 
expressly says that throughout our abstract reasoning 
we must keep in our mind the concrete picturable 
background. We need not inquire whether it is a 
physical body with other properties besides the merely 
spatial, but we do require a body qua solid and shaped, 
and must never lose sight of it. All we require to 
keep clearly in mind is a spatial embodiment of the 
magnitudes; but we cannot dispense with that. And we 
must always interpret the abstracted features by refer- 
ence to the concrete, picturable whole from which we 
have abstracted them. 'Figure' will be the pictured 
solid thing considered purely so far as it has shape. 
'Line' will be length, not without breadth (in the sense 
of excluding breadth), but the pictured solid conceived 
in abstraction with reference only to its length and 

1 Ibid. 

2 Reg. xiV, p. 452. 

so likewise with 'point', 'plane' and the like. So also 
'number' will be the object measurable by multiplicity 
of units. But we must never lose sight of the pictured 
something whose multiplicity these units express 
though provisionally we disregard its other proper- 
ties. 1 

Descartes accuses the arithmeticians and geometers 
of confusion of thought. The arithmetician tends to 
regard numbers as abstracted from every material 
thing, yet as having a kind of isolable, picturable exist- 
ence, whereas they are separable only as a result of his 
abstracting. So the geometer, having first regarded the 
line in abstraction as length without breadth, and 
the plane as area without depth, forgets that these are 
mere modes abstracted features not isolable ele- 
ments of bodies. He proceeds to generate plane from 
line, an operation which pre-supposes that the line 
from which the plane flows must itself be body, where- 
as line proper is merely an abstracted mode of body. 

Is Descartes's theory of vera mathesis then value- 
less? Rather is it the case that here he is still feeling 
his way and his account of it is a somewhat blundering 
and roundabout mixture of several different ideas. 
At all events, there can be no doubt about his math- 
ematical discoveries, which he is here trying to ex- 
plain, and he can be better understood by reference to 
those features which reappear in the Dfscours. 2 Des- 
cartes reformed the contemporary algebra by introduc- 
ing an improved, easy and consistent notation, so that 
he was able to reformulate problems about magnitudes 
in terms of proportion in general. He then conceived 

1 Reg. xlv, pp. 442-6. 

J Cp. Liard, op. cit., pp. 35-53. 


the brilliant idea of calling geometry to the aid of 
algebra calling the imagination to the aid of the 
intellect. Descartes's project was, accordingly, the 
graphic solution of equations. The third book of La 
Geometrie 1 is devoted to this task. Notwithstanding 
its title, however, La Geometrie is not geometry in the 
common meaning of the word, but vera mathesis (i.e. 
algebra illuminated by an appeal to spatial intuition). 
In the course of his work on the graphic solution of 
equations, 2 he was led to the discovery of the analytic 
or co-ordinate geometry. This is not the new universal 
mathematics itself, but the result obtained by reversing 
the procedure of the new science. The fundamental 
idea of the vera mathesis is the solution of problems 
expressed in algebraical terms by means of geometrical 
figures; but the analytical geometry is based on the 
idea of substituting for the spatial figure an algebraical 
formula which gives the law of the generation of the 
figure; for instance, that constituting the equation 
which prescribes the successive positions in a plane 
through which a point flows as it constitutes any 
required visible straight line. 

Interesting as it may be, however, to observe the 
nature of Descartes's mathematical discovery and to 
find out what its relation was to his vera mathesis, 
the important question is: Does his vera mathesis 
turn the edge of the criticism threatening his method ? 
Even the immature account given in the Regulae 
makes it clear that the vera mathesis assumes contin- 
unity continuous magnitude and assumes it is an 
object, not of the pure intellect, but only of the imag- 

1 Published with the Discours as a specimen of the new method. 

2 Cp. Liard, loc. cit. 

ination. Also, the Regulae, more clearly than any other 
of Descartes's works, betrays the desperate inadequacy 
of his theory of the imagination, and the utter failure 
of the Cartesian method in explaining anything so 
concrete as imaginative knowledge or experience. 

(i) In the first place, the method is an analysis, which 
proceeds by subtraction, together with a synthesis, 
which proceeds by addition. Therefore, a mind which 
reasons in accordance with this method, can deal with 
nothing that is not an aggregate of simples. Thus the 
only field of facts which it could hope to explain 
(without explaining away) seems to be that of numbers 
and numerical proportions. For everywhere else the 
intellect would be confronted with wholes concretely 
or genuinely one, unities which are not units or sums 
or assemblages of units. So the intellect, as Descartes 
conceives it, would necessarily fail to achieve know- 
ledge, even in the spheres of the special sciences such 
as geometry, dynamics, physics and the sciences of life 
anywhere where we come into contact with contin- 
uity, motion, and the like and a fortiori in the field 
of philosophy. 

What is here being urged against the method is 
precisely its abstract formulation of the principles con- 
trolling mathematical reasoning at its best; and Des- 
cartes claims to have shown that everything in the 
universe falls within the grasp of mathematics; except 
the facts, activities and achievements of self-conscious 
mind. But our examination shows that the chief bur- 
den of the task of mathematics falls on the imagination 
and not on the intellect. The intellect cannot (or at 
any rate does not) explain continuous magnitude, but, 
on the contrary, it borrows fron the imagination the 


pictured thing in order to throw light upon its own 
abstract procedure. Even when it is a question of dis- 
cussing number, we are expressly enjoined to keep the 
reasoning of the intellect within the control of the 
imagination by reference to the concrete res numerata. 
Though he says that it is always possible to reduce (at 
least in part) a continuous magnitude to an aggregate, 
by the introduction of an assumed (fictitious) unit, this 
is no more than a legitimate device for facilitating 
measurement. But there is no evidence that Descartes 
thought and he certainly does not try to prove that 
the continuity of the magnitude is, by this device, 
explained : that is, shown to be a sum of discrete and 
simple units (like the postulate of an infinite juxta- 
position of points in a line, which it is also convenient 
to suppose for certain purposes). 

(ii) Secondly, the exposition in the Regulae brings 
out the defects in Descartes's account of the imagina- 
tion. These are the consequences of the purely abstract- 
ing and eliminating analysis which predominates in 
Descartes's thought (such synthesis as he contemplates 
being only the adding of determinants). He applies an 
analysis of this kind to the contents of various forms of 
experience, such as sense-perception, imagination and 
memory, as well as of mathematical, scientific and 
philosophical reasoning. He detects in all an abstractly 
identical common feature something to be known by 
a pure, undifferentiated, always identical vis cognos- 
cens. So he says that one and the same power is at 
work in sense, imagination and reason, and that it is 
the instrument of knowledge in all fields of investiga- 
tion. The differences which distinguish these various 

experiences from one another arise solely from differ- 

ences in the objects to which the undifferentiated vis 
cognoscens is applied. 1 So the total experiences are not 
different forms of knowledge; there is only one form of 
knowledge: i.e. the intellect, functioning purely and 
alone the vis cognoscens applied to ideas that are in 
the intellect itself. Sense and imagination are not 
knowledge, though they include knowledge (i.e. they 
include the effects of the absolutely identical func- 
tioning of the vis cognoscens), as one element in 
them. But they include, in addition, changes of states 
(or shape) in certain organs of the body which are the 
objects of this element of knowledge or on which it 
casts its light. The eliminative analysis has thus re- 
duced these apparently different modes of conscious- 
ness, each of which is a total or concrete experience, 
to a single undifferentiated power of knowing a 
purely spiritual awareness together with various 
bodily changes, to which that awareness is directed, 
or of which the spiritual power is aware. 

It does not seem to occur to Descartes, at any rate in 
the Regulae, that any further explanation of sense- 
perception or imagination is required. He speaks of the 
vis cognoscens receiving shapes from the sensus com- 
munis or the phantasia; it is said to see, touch, etc. 
when it applies itself to common sense and imagina- 
tion. So the vis cognoscens in imagination apprehends, 
irt the central organ of sense, at times a sense impres- 
sion (aesthema), and at times a phantasma a survival, 
or record, of similar impressions apprehended in the 
past. In the first case it is sense-perception, in the 
second imagination (phantasia); and somehow we are 
able to recognize the former as the effect of an ex- 
1 Cp. Reg. xii, pp. 415-6. 

G 97 

ternal cause on the peripheral organ of sense that 
is, we perceive the external thing. 

Imagination here is the visualisation of certain 
shapes and figures in the bodily organ of imagination 
(phantasid), which is a 'vera pars corporis', and the 
shapes and figures are copies of the shapes of the outer 
bodies. In apprehending a visual idea of a spatial figure, 
therefore, the vis cognoscens is, apparently, apprehen- 
ding the shapes and mutual relations of the parts of the 
bodily organ of imagination, and, as this is an exact 
reproduction in miniature of things in the external 
world, we can (apparently) be confident that we are 
apprehending true models of the external things. But 
no meaning whatever can be attached to this descrip- 
tion of sense-perception and spatial imagination, ex- 
cept on the assumption that the percipient knows the 
external causes of his imaginations (i.e. the shapes and 
inter-relation of parts of the outer bodies) independen- 
tly of sense-perception and imagination. 

But Descartes says that the vis cognoscens alone can 
know, and that it apprehends only its own ideas or 
changes in the bodily organ; and again 1 he speaks of 
imagining as a function in which we use the intellect, 
not in its purity, but assisted by the forms depicted in 
the phantasia. How is such a use possible on Descartes 
theory, unless we postulate a second vis cognoscens 
which, by employing the first vis cognoscens while 
that is apprehending the images in the phantasia, 
knows the external things? 

So no results from the vera mathesis, and no results 
from his interesting theory of the physical world, can 
invalidate our criticism of his method. Vera mathesis 

1 Reg. xfv, p. 440, 1. 2$f. 

is not a result of the activity of pure intellect according 
only to the rules of the method. Yet how can it ever 
be 'assisted' by images? Descartes must be feeling after 
some more concrete form of thinking and knowing, 
though he fails to reach it in the Regulae. If he had, he 
would have been forced to adopt so radical a modifica- 
tion of his theory of method as to be tantamount to 
abandonment of it. 




The critic who objects to Descartes's theory of the 
intellect and of method on the ground that the true 
mode of reasoning is not necessarily of this kind, will 
be challenged to show that any method of reasoning 
exists, or can be conceived, other than analysis into 
simples followed by synthesis into aggregates. If there 
is any mode of analysis which can resolve a concrete 
fact without disintegrating it which can do anything 
other than split it up into simple natures what is it? 
At least we should be able to show it at work. 

This challenge raises a two-fold issue : (a) Are there 
any concrete facts, or wholes, of the kind alleged, that 
are recalcitrant to Descartes's analysis and synthesis? 
Is there anything in the universe, anything in the field 
of experience, except simples (units) and sums, linUs 
and chains? Is anything one, which is not either a 
simple element or a sum a plurality of simple ele- 
ments conjoined a network of relations covering and 
comprehending the many, without penetrating or 
affecting their single natures? (b) If we assume that 
there really are such recalcitrant, concrete wholes 


not merely that there are objects which seem to be 
such because of a confusion in our limited knowledge 
then in what sense, if in any, are such facts ex- 
plicable or intelligible? What mode of reasoning is 
available to throw light upon a fact which is such that 
nobody, however patient he may be, could analyse it, 
in that sense of 'analyse* in which it means 'resolve 
into simple constituents that are separate and separ- 
able, from one another and from the whole; that are 
externally connected and separately conceived'? 

As to (a), there is no doubt that we do commonly 
attribute a variety of modes of unity and wholeness 
to the objects of our experience. We do take it for 
granted that a many may exhibit different types of 
connectedness or cohesion; and also that a unity, or 
one of^many (a complex) may differ from other unities 
or complexes in the kind of its one-ness, the type of its 
unity, wholeness or compoundedness. We shall give 
a general sketch of our 'ordinary' views on such mat- 
ters, enumerating three main types of unity or whole- 
ness, though some of them the plain man may repud- 
iate and others he would not distinguish so rigidly or 
so dogmatically. 

(i) To begin with there are wholes (though so ap- 
plied the term is used vaguely) which consist of parts 
and are resoluble into them. Such a whole consists of 
p'arts separable from it and from one another, in 
existence and character, in being and intelligibility. 
For example, a square contains two right-angled trian- 
gles (potentially, if not actually); the number 6 consists 
of 3 and 2 and i, or of six units or two 3*5. Within 6, it 
may be said, there are the same 2 and 3 and i as can be 
conceived in isolation or may be found in some other 


containing number (12 or 24). In the square there are 
the same triangles as result from its division, or may 
be conceived in isolation, without ever having formed 
it. But these examples are open to dispute. Let us, then, 
take material things such as a wall or a watch. These 
may be taken to pieces and then put together again so 
as to reconstitute their respective wholes. Their vari- 
ous parts compose and are contained within them; but 
all of them may also enjoy free existence, or enter into 
other wholes, individually and even specifically differ- 
ent. Within their wholes, no doubt, the parts are relat- 
ed in certain ways and adjusted to one another in 
accordance with a certain arrangement or plan. But 
however essential the plan may be for the being and 
conceivability of the wholes, it is external to and sits 
loose upon the parts. The buttons of my coat ^re not 
altered whether they happen to be on the coat or off it 
(unless some accident disintegrates them). 

(ii) In another kind of whole the relation of the parts 
to the whole is, at least for common opinion, in dis- 
pute: the chemical compound and its constituents. 
Oxygen and hydrogen do not seem to be the constitu- 
ents of water in the same sense as bricks are con- 
stituents of a wall. It is true that they are isolable in 
so far as they can be recovered out of water by chemi- 
cal analysis, but are they present in water in the same 
way as bricks are in the wall? Or have they been 
absorbed, merged into the genesis of water ? 

What of the principle of the conservation of mat- 
ter ? One may retain one's belief in that and yet deny 
that it is relevant to the present issue. The principle 
asserts that in all chemical changes something called 
'matter* is not increased or diminished, but conserved; 


but this something (in the case of the formation of 
water) is certainly not oxygen or hydrogen. It looks as 
though neither oxygen nor hydrogen nor yet water 
persists throughout the change we call combination, 
otherwise there would be no change or coming into 
being. If the oxygen and hydrogen persist, where is the 
change? and the water was not there at the begin- 
ning. Still, it may be objected, this ignores what is 
really the important question: Is there no sense in 
which water may truly be said to have been, even 
before its perceptible emergence ? What kind of modi- 
fication to the constituents is necessary for a chemical 
change? May it not be such that they can undergo it 
and nevertheless persist? A scientific theory answers 
this question. Oxygen and hydrogen certainly are diff- 
erent from water, but the difference is only secondary 
only in the derivative perceptual qualities and is 
dependent upon persistent identity of substance. In 
chemical change the atoms are reshuffled; so to speak, 
they dance to a new tune, or are newly grouped. 
Water is a sort of mosaic of oxygen and hydrogen 
atoms with a very definite pattern, and this new 
arrangement is the basis of the new qualities. So the 
same atoms, which, moving freely in isolation, were 
the substantial basis of the characters we perceive in 
the gases, are now the basis of qualities we perceive 
in the water. The orthodox chemist still holds this 
theory, or some refinement of it in which 'equilibrium 
of electrical charges' (or the like) is substituted for 

But, even if we assume this to be true, what does the 
theory assert as to the relation of the chemical consti- 
tuents tp their compound? The atoms of hydrogen and 


oxygen are grouped in a determinate fashion/ and 
when so grouped they display new qualities; so the 
grouping has essentially altered the atoms. As grouped 
atoms behave quite differently, it is absurd to suggest 
that the grouping is a mere external arrangement. No 
isolated atom would dream of so behaving. Not just 
atoms, but grouped atoms, are the basis of the qualities 
of water, and it is useless to attribute these qualities, 
as they appear in water, to the atoms by themselves. 
Nor is it possible to explain the differences literally by 
using the antitheses of primary and secondary quali- 
ties, or substance and accident. The grouping is not 
analogous to the arrangement of bricks in a wall or 
buttons on a coat. In other words, a chemical com- 
pound differs from an aggregate in the nature of its 
wholeness. Here the combinables exhibit a mode of 
cohesion inter se which radically affects their existence 
and character. 

It would be agreed that, in both of these two kinds 
of whole, the parts are original, primary and simple, 
and the wholes derivative, secondary and complex. 
Light is therefore, thrown on the aggregate or com- 
pound by analysis of it into its constituents and sub- 
sequent recombination. Difficulties arise, however, 
when one looks closely into the supposed analysis. 
If the aggregate were really no more than has been 
described above, it would not be one or whole at alj. 
Each of its constituents is one, but no unity other than 
this is allowable in terms of the description given. 
There is no coherence of the simples; they are to- 
gether, but that is how we regard them and not a way 
in which they are. So, strictly speaking, there is no 
1 Or 'pattern'?. (Ed.). 

whole and the elements are not constituents; and since 
the aggregate is not a proper whole, it does not admit 
of analysis. What is called analysis is simply the pick- 
ing out, one by one, of the grains of the heap. It is 
substituting the clear conception of many singulars 
for a confused and mistaken impression of wholeness. 

On the other hand, it is to be noted, none of the 
examples given really fulfils the requirements of the 
description. They are not precisely and accurately 
aggregates in the sense required. A watch is a whole, 
but its wholeness is derived from the purpose embodied 
in it. Only as a variety subordinate to plan and as 
means to a common end are the 'parts' (the spring, the 
wheels, and so forth) parts of the watch properly so- 
called. But when so considered they are obviously not 
isolated nor isolable, nor are they severally intelligible, 
nor capable of separate existence or description. What 
may seem to be a mere collocation carries with it real 
effects in the collocated parts, which are not less real 
or present because difficult to trace. For example, 
bricks in various shapes of walls or in different 
positions in the wall suffer various strains and stresses; 
and so with pieces of wood when worked into a chair. 

In the case of the chemical compound there are 
similar difficulties. No doubt the combinables are orig- 
inal and primary and the compound results from their 
self-sacrificing coalition. But it is, at best extremely 
doubtful whether the properties of the compound can 
be deduced from or elucidated by the properties of the 
combinables. Here there is scope for analysis, but 
could the analysis retain or explain the whole could 
it throw light on its character ? In the aggregate what 
>vas corifused and complex was rendered, by analysis, 


clear and simple; but this was not really analysis, for 
there was strictly nothing to analyse. In the compound 
there is a whole to be analysed and traced back to its 
elements, but when this has been done we are left with 
the character of the compound as unintelligible as 

However, let us ignore these difficulties here, as they 
usually are ignored. Let us admit that there is a charac- 
ter of wholeness, both in aggregates and in compounds, 
which the statement of them in terms of their constitu- 
ents does not touch, and that 'analysis' is an inapprop- 
riate name to give the procedure which we call 
explaining. Still, the procedure retains a certain value. 
We have got things clearer by getting at the simpler 
elements; the constituents and combinables are con- 
ceivable definitely, and in substituting them for the 
wholes we do seem to have made some advance in 
knowledge. We learn by it but less than we com- 
monly suppose, and not quite that which we com- 
monly suppose. 

(iii) But there are also wholes of quite a different type 
the elements of which are not isolable 'concrete* 
wholes or concrete facts, whose parts are only constit- 
uent moments. Properly speaking, such wholes do not 
consist of parts and there are no isolable elements from 
which they are derived. Here the whole is original 
and substantial and the 'parts' are derivative and adjec- 
tival. The whole differentiates itself; it is not the parts 
which, enjoying at first each its own separate being, 
combine to form or are adjusted to constitute the 
whole. The parts are not separable even in thought 
not even intelligible apart from consideration in terms 
of the whole. Yet, though this is so, if we wish to 

understand the whole we are forced to distinguish 
'parts' within it and recognize them in their differ- 
ences as essential to its being and to its characteristic 
mode of oneness. 

To maintain that such wholes exist is not easy. How 
can we defend the notion of a whole whose parts have 
no character or existence except as constituting the 
whole? Can a whole have its unity essentially in 
variety ? Locke cannot understand how this can be and 
denies its possibility (though he begs the question by 
admitting that it is 'made up of its parts). To say that 
its unity is of the essence of its diversity, and its diver- 
sity of the essence of its unity, seems preposterous. It 
seems like saying that a thing is black and white all 
through and is each because it is the other. 

Nevertheless, the kind of fact of which we are think- 
ing is whole in one sense at least: it is genuinely 
single or one not abstractly, as a unit or a simple 
quality, but concretely. That is, its unity, though it is 
continuous and is not resoluble into elements and 
their connections, is not a monotone. There are differ- 
ences, there is articulation; and, when we reflect on 
what, in this sense, is a concrete whole, we must recog- 
nize in it this diiferentiation or diversity, and the 
diversity as essential to its organization or wholeness. 

Although such concrete facts are, beyond question, 
real and though they are usually recognized, there 
is a marked deficiency of appropriate terminology for 
their description. It seems natural and easy for the 
mind to function (roughly) in the Cartesian manner, 
and in everyday life we tend to use terms which inter- 
pret wholes of this nature in Descartes's way, so long 
as our thought is relatively effortless and careless. The 


terms commonly used to mark the non-isolable 'parts* 
of such concrete wholes are 'features', 'aspects', 
'organs', 'members'. Some, like 'aspects', are apt to 
suggest that the unity is really abstract and monoton- 
ous though it looks concrete and diversified; others, 
like 'members', 'organs', etc., tend to suggest that the 
parts are isolable to some extent like the constituents 
and combinables discussed above. Accordingly, con- 
crete wholes of this kind may seem to be more ap- 
parent than real. But this is so, not because the parts 
are unreal or the wholeness imputed, but because the 
terms in which we describe them are unsatisfactory, 
or difficult, or in other respects subject to criticism. 
All that one can do is to select the least inappropriate 
of the current terms, while remembering alway^ their 
^inadequacy; so one may try to retain mastery of one's 
own terminology, and not use terms thoughtlessly so 
as to accept unconsciously their misleading associa- 

Prima facie at any rate there are several varieties of 
wholes with inseparable parts. Two stand out as espec- 
ially typical, but they seem to differ specifically, 
though further consideration might show that there 
are grounds for suspecting that not even these are 
genuinely irreducible types or species that we are 
dealing here not with specifications of a genus but 
with variations on a theme. But we shall, in either 
case, find it convenient for the present to have general 
terms to mark the theme or genus. We shall call all 
such wholes 'concrete unities', and their parts that 

is, any diversity of the kind that belongs to such 
1 08 

wholes 'moments' (or 'constitutive moments'). 1 We 
do not insist on the terms. They have been used com- 
monly enough in some such sense in philosophy and 
they are not altogether inappropriate. 

(a) Of the two types of concrete fact we are to 
consider, the first is a living organism. We tend to form 
different conceptions of a living organism, some less, 
and some more, adequate, and, therefore, likewise of 
its inseparable parts. We regard it as an individual 
whole an equilibrium of vital activities or a concrete- 
ly single cycle of such activities (the parts or moments 

1 The Latin word 'momentum' has (among others) the meaning of 
'a decisive factor'* e.g. the last straw breaking the camel's 
back. This perhaps suggests that it is an isolable constituent. Is 
not the last straw, an objector may ask, a separable increment 
to an t existing burden? It is, however, only qua last that the 
straw breaks the camel's back, and it is only qua last that it 
merits the name of decisive factor, 'momentum' that is, only 
as inseparably one with and completing the given burden. Given 
the total weight, any straw might be called 'momentum'; but 
no straw per se, qua isolable, deserves the title. The German 
word Moment is often used with the meaning required. In the 
analysis of involuntary movement, for instance, a German 
would distinguish das physiohgische Moment from das psy- 
chologische Moment two factors, each of which makes a dis- 
tinct and indispensable contribution to a concrete fact, which 
is a change indissolubly single, bodily and psychical in one; as 
a curve is concave and convex at once and yet a single and 
"indivisible direction. (Cp. Aristotle, Nicomachean Ethics, 1102 
aji.) If we are to take even the first step in understanding in- 
voluntary movement, we must take into account the two 
'moments', but though the distinction is in no sense arbitrary, or 
subjective, yet the 'moments' revealed have no separable exist- 
ence, and it is misleading to speak of them as 'factors'. 
Lewis and Short give, as the second meaning of 'momentum' : 
'A particle sufficient to turn the scales'. (Ed.). 


of the cycle are the various subordinate vital processes, 
such as respiration, reproduction, etc.), or else as a 
living and active federation of cells, or again as an 
immanently teleological system of co-operating or- 
gans, which in their functioning are both means and 
ends not only to one another but also, in a sense, to 
the whole. 

It matters little which view we adopt, we are clearly 
in each case differentiating the whole, from which we 
start. Yet, in differentiating, we are eo ipso integrating 
it. We are not putting it together out of isolable 
constituents nor deriving it, by combination, from 
combinables in themselves separate. All subordinate 
activities (such as breathing, digesting, etc.) all federat- 
ed colonies of cells, all co-operating organs contribute 
by their differences to the being and maintenance of 
the single, individual life or federal unitary policy 
or unitary (though far from monotonous) co-operative 
work. At the same time, however, these articulations, 
while they contribute to the whole, depend upon it for 
their existence and nature, their being and intelligibil- 
ity. In the whole, and only in it, they live and move 
and have their being, Sever them in fact from the 
whole and they cease to be; sever them in thought, and 
they cannot be conceived or described adequately. On 
the other hand, the unity of the whole is, all the time, 
nothing but the conspiracy of their differences, tHe 
equilibrium of their co-operation, and its individual 
life is the cyclical movement of which they are the 
constitutive moments. Yet the inseparable organs of 
the living organism, though moments and not isolable 
parts, do seem to be, in some sense, the integrants of 
the whole, and so we may call them its articulations. 


There are subordinate systems of activity within each 
system, and colonies of cells within each federation, as 
well as individual cells within each colony. 

There is thus one typical variety of a concrete fact, 
of which a living organism and its articulations may 
be taken as a conspicuous example. This seems to be 
marked off from the second variety 1 because the 
moments of the living organism, in constituting the 
whole, are (or seem to be), in some sense, integrants of 
it, which is not the case with the other kind. If, how- 
ever, you ask by whom the organism would be taken 
as a conspicuous example, the answer must be : An ob- 
server with some scientific education and perhaps 
some philosophical education. The ordinary untrained 
observer commonly, not only regards as parts of the 
organism what are isolable (e.g. teeth), but continues 
so to think of them even when they are isolated. Such 
an observer would realize very imperfectly that, for 
example, an eye or a hand, which in living and func- 
tioning are genuine parts or articulations of the organ- 
ism cease eo ipso when separated from it, to be what, 
by an abuse of language, they are generally called. A 
'dead hand' is strictly a contradiction in terms. 
Though we are ready enough to speak of the organism 
in terms of cells, organs, and the like, we usually 
separate, in thought, the cells from their life, the 
organs from their functions and processes from the 
systems which carry them; and we tend to erect these 
abstracted organs, cells and structures into substantial 
things with characters which they are supposed to re- 
tain, whether the whole to which they belong is living 
or dead. 

1 Vide (b) below, pp. ii2ff. 

The view that such abstraction distorts the trutl 
would probably be accepted at first by the scientist 
but he would set it aside as useless for the actua 
detailed work of his science. He would say that w< 
must assume the separateness of the parts for the spec 
ial purposes of the science. They compel him to ab 
stract and pay attention to the facts of the separate 
behaviour of the different elements and the laws whicl 
govern this behaviour. So, he would say, even in th< 
organism, science must start from the simple and pro 
ceed to the complex, and must go on the assumptioi 
that patient investigation will detect simple constitu 
ents and discover the laws of their behaviour, whethej 
in or out of the living whole, so that in the end we car 
explain the organism as a complex or composite result 
as if it were an aggregate. 1 

(b) The second variety of concrete facts may b( 
called spiritual wholes, of which knowledge (or truth) 
beauty (or aesthetic experience), goodness (or mora 
experience) are examples. The moments here are dis 
tincta distinguishables into which the whole neces 
sarily differentiates itself in philosophical analysis 
Since, without such analysis, spiritual reals cannot be 
properly understood, the distincta are necessary to it< 
intelligible being its real or essential being. It is con 
venient, therefore, to call them its 'implicates'. Bui 
they have not co-operated to form it; nor do they exist 

1 For examples to the contrary, cp. L. von Bertalanffy, Moderr 
Theories of Development (Oxford, 1933) and Problems of Lilt 
(London 1952); Joseph Needharn, Order and Life (Cambridge 
1936) and J. S. Haldane, The Philosophical Basis of Biology 
(London 1931), Organism and Environment (Yale Univ. Press 
1917). (Ed.). 

or occur, either in isolation outside it, or separably 
or even inseparably within it. Their being is no more 
than their emergence under that philosophical analysis 
which alone shows what their spiritual whole implies 
what in truth it is, and so what they (its moments) are. 
So these 'implicates', these constitutive moments of a 
spiritual real, are precisely what reflection on that real 
shows it to imply. 

Here we may meet an objection. No doubt, know- 
ledge and truth, beauty and goodness are familiar 
omni-present realities of human experience; no doubt 
they do embody and express feelings and activities of 
the spirit; no doubt they are feeling, will and thought 
realized and objective. Therefore it is reasonable to call 
them spiritual experiences. There really is a moral 
order, a kingdom of ends, sustained by and embodying 
the effective will for good of moral agents; and within 
each whole there are smaller spiritual wholes, also real 
e.g. morally good institutions, and characters, and 
acts. And there are real aesthetic experiences, and 
there really is knowledge experiences in which we 
possess or are possessed by truth. But there is nothing 
to show that these realities are the sort of wholes (with 
inseparable parts) that we are supposing. If they were, 
they could not be analysed. And if they were such 
wholes and could be analysed, there is nothing to show 
that this procedure of analysis would elucidate them or 
help us to understand them. 

There is nothing to show all this, except philosophy. 
And, if more closely examined, the concrete facts of 
the first type would probably lose their apparent 
difference from those of the second. All such facts are 
the proper objects of philosophical study, with them 

H 113 

and with little else philosophy has to do. And what the 
philosopher does with them is to try to elucidate them 
by the very procedure which the above objection 
declares to be impossible or worthless. He analyses 
them, not into constituents, but into implicates. Such 
analysis would be, eo ipso, synthetis into a concrete 
whole which is real, because it would at once exhibit 
the fact constituted by the mutually implicated 
moments as single in a unique way, and as varied in a 
unique way. And since this is necessary for the discov- 
ery of the implicates, the philosophical analysis would 
elucidate the whole. 

But there is a possible misunderstanding of the 
nature of implicates which must be avoided. Consider 
the earth's revolution round the sun. That may be 
resolved, mathematically, into two component move- 
ments : one in a straight line and one towards the sun's 
centre. Neither is actual nor a constituent of the 
earth's movement. Neither makes an actual contribu- 
tion to the revolution. The mathematical statement 
does not say that they in fact co-operate to produce it, 
nor that they are now integrants of it, but just that the 
movement takes place as if it were a compromise 
between them. Again, the numbers 4 or 6 do not con- 
tain units, nor does the square contain lines or tri- 
angles. 3 and i need not, in fact, co-operate to produce 
4, nor be components of the whole 4. They are reached 
by analysis which destroys the whole, but the whole 
does not contain or consist of these factors. Now it 
may be thought that 3 and i are implicates of 4; or i , 
2 and 3 implicates of 6; or two right angled triangles 
implicates of the square. It may be said that in such 
examples we have clear distincta with separable as- 

pects or moments but not separable parts, and as these 
moments are not integrants of the whole they seem 
to accord with out definition of 'implicate'. But these 
mathematical examples fail to exemplify the proper 
conception of implicates in a spiritual real. For these 
analyses are not the only possible analyses. Each gives 
only one possible alternative set out of many possible 
sets of moments which may be distinguished by reflec- 
tion on the whole. 6 may, but need not, be analysed 
into 2, 3 and i; and we may suppose other sets of 
rectilinear figures that will together constitute a square 
But, in order that the moments should be true implic- 
ates, we must, for example, be able to show (a) that the 
earth's revolution requires precisely these and no other 
motions; and (b) that these motions can neither be nor 
be conceived except as constituting that particular 

It is characteristic of most philosophical theories 
that they are concerned with concrete facts and try to 
explain them by analysing them into their moments 
and (eo ipso) synthesizing them into wholes. But not 
all philosophers recognize expressly that such wholes 
are their proper subject and such analysis and synthe- 
sis their proper method. Hegel does so uniformly and 
consistently; Kant does so in the main, notwithstan- 
ding lapses and inconsistencies of detail. But many, 
who do not expressly recognize such wholes, and some 
who even repudiate them, are, in their actual specula- 
tions concerned with nothing else. 1 Many philosophers 
(especially Kant and Aristotle) will sometimes them- 
selves profess to be seeking constituents when they are 
clearly looking for implicates or what would be im- 

1 Cp. Plato, Theaetetus 2O4A, and Leibniz, Monadology, 1-2. 

plicates if their arguments were sound (e.g. Aristotle's 
conception of wpolr^ vhq, zffiog and (rripi>]<rt$ 
as constituents of body as such). 1 Let us consider two 
examples : 

(i) Leibniz opens the Monaldology with what seems 
to be a quite uncompromising repudiation of wholes 
with inseparable parts. He takes it for granted, appar- 
ently, that everything in the universe is simple or com- 
pounded of simples. A whole is an aggregate or com- 
pound with simple parts, and 'simple' means 'with- 
out parts', while, apparently, all that is not simple 
has parts. But the simples are 'monads', each a unique- 
ly individual spiritual real, characterised by uniquely 
individual 'appetitions' and 'perceptions' which are its 
implicates. Its appetitions are tendencies to unroll its 
own series of conditions, and its perceptions ar those 
expressions of many things in one from and to which 
the monad passes in its successive phases. The monad 
is one without parts because, unlike an extended 
whole, it has no isolable constituents. Yet its unity, 
says Leibniz, requires variety, both coexistent and 
successive, for it maintains its simple being in, and by 
virtue of its successive phases. It is a many expressed 
in one. It is simple concretely (that is, in and by virtue 
of its simultaneous internal variety); yet at each phase 
it is a different many expressed in one, and it maintains 
itself by being successive. Each monad is this and no 
other, because its appetitions and perceptions are these 
and no others because they are uniquely graded both 
in their intensity and their distinctness. And they are 
uniquely so, because they are this monad's special 
1 Contrast, however, Physics 19137-12, and De Generatione et 

Corrupt/one, 329332-35. 


phases and tendencies to change. In other words, they 
are its implicates its special detail which this 
monad selects out of infinitely various differences as 
its especial implicates. It, and it alone, is their unroll- 
ing, and at each successive phase they are its unrolling. 
In a later paragraph, 1 he says that, just as a town 
looks different from different points of view, so the 
universe, though itself single, is made up of various 
aspects, or views of itself (i.e. monads). Each such as- 
pect or view is the special relation to the whole of each 
monad in its unique and detailed perspective. This 
comparison, no doubt, is helpful and reliable, up to a 
point, and is legitimate with the reservations that 
Leibniz himself adds. For, strictly according to his ter- 
minology, the only reals in Leibniz's philosophy are the 
simple substances and their states (their perceptions 
and appetitions). The singleness of the universe, there- 
fore, must be conceived as a qualitative completeness 
or intensive fulness of the spiritual life which is infin- 
itely graded into the scale of monads. The infinite de- 
tail of the universe, on the other hand, is the multitu- 
dinous aspects which are the monads the hierarchical 
system or articulated scale of infinitely graded monads. 
The total energy and life is immanent in every grada- 
tion of itself. Each monad enfolds the infinitely various 
detail, but each expresses or mirrors this infinite vari- 
5ty, this intensive fulness, which is the whole, at its 
own uniquely limited degree of intensity. Because 
every monad is thus an articulation of the total spirit- 
ual life, and because the implicates of that life constit- 
ute each in its individual degree because that is so, 
therefore each change and passing condition of any 

1 Monadology, 57. 


one monad eo ipso involves a corresponding change 
in every other (the pre-established harmony). Hence, 
the self-containedness of each monad, so far from ex- 
cluding, necessarily pre-supposes the adjustment of 
each to all. 1 

(ii) Kant's central task in The Critique of ?ure 
Reason is the analysis of fact into its implicates. Fact 
is something essentially known or knowable by any 
intelligent being. That this is so necessarily implies a 
spiritual whole, which Kant calls 'experience' (but 
which might better be termed 'knowledge-or-truth'). 
This differentiates itself, under Kant's analysis, into 
two correlative articulations: (a) the self-conscious, 
scientific mind, and (b) its correlative, the correspond- 
ingly organized object of mind - the ordered world of 
physical science. Reflecting on this whole, and its two 
differentiations, Kant further resolves each of them 
into implicates. These co-implicates of both articula- 
tions of the spiritual reality, knowledge-or-truth, are 
the implicates of any and every fact. They appear as 
(e.g.) 'the manifold of sense', 'the forms of intuition', 
'the schematized categories'. 

If this rough sketch does represent the main drift of 
Kant's teaching, there is a tragic perversity in his ex- 
position. He constantly describes implicates as though 
they were constituents. He talks of the forms of 
intuition and the categories as 'elements' of knowledge 

1 Cp. R. Latta, Leibniz: The Monadology, esp pp. 108 ff and 2oo- 
202, and B. Russell, The Philosophy of Leibniz, esp. Chs. XI and 

found in us a priori. 1 As a result he substitutes a des- 
cription of what pretend to be stages of knowledge for 
the critical analysis he is plainly aiming at. He is, 
partly at least, responsible for the common misinter- 
pretation of his doctrine as analysing mind into con- 
stituent parts the various faculties like the bits of a 
machine. By the same misinterpretation, the world is 
represented as a manifold datum, plus connections, or 
with arrangements, introduced into it, or superimpos- 
ed on it by the mind. According to this misinterpreta- 
tion, the manifold is at first passively received, and 
then, step by step, organized into a systematic body of 
knowledge by spontaneous activities of the mind (intu- 
ition, imagination, judgement, etc., in succession). 
We may summarize our argument, then, as follows : 
(i) The proper subjects of philosophical study are 
concrete facts. These are unities, but neither units nor 
aggregates. They are wholes, but neither complex nor 
compound. They are wholes with inseparable parts 
wholes which determine, and are determined by, two 
or more implicated moments. There seem prima facie 
to be at least two more or less irreducible kinds of 
such facts: organic wholes with integrant parts, or 
differentiations and spiritual reals with implicates 
wholes which imply and are implied by distincta that 
constitute but are not integrant of them. It is to be 
suspected, however, that, on more careful considera- 
tion, organic wholes would turn out to be disguised 
and imperfectly analysed examples of spiritual reals. 

1 Krit. der Reinen Vernunft. B 166. Cp. in Prolegomena to any 
Future Metaphysics 18, the gratuitously introduced distinction 
between 'judgements of sense-perception' and 'judgements of 


(ii) The proper philosophical method is analysis, not 
into constituents, but into implicates or moments, and 
this is eo ipso synthesis. This analytic synthesis or syn- 
thetic analysis makes clear how the unity is concrete 
that is, it shows the unity as an intelligible union 
of an intelligible variety. A two-edged process of this 
kind is the only adequate treatment of such matters as 
knowledge-or-truth, goodness and beauty in short, 
any problem in philosophy : in logic, metaphysics, 
morals, politics or aesthetics. But the power to treat 
any problem thus, or to appreciate such a process of 
analysis, presupposes in the student a long and patient 
apprenticeship. The student must gradually work up 
to it through the lower levels of investigation, by a 
progress in the course of which he tests, remodels, 
cancels and recasts many erroneous theories of his 

, Consider, for example, Plato's account of the origin 
of the state in the Republic, and observe the genuine 
and vital necessity for the earlier and imperfect analy- 
ses by which he leads up to his own mature and con- 
sidered theory. Civilised society is not the result of 
mere contract between separate persons; not the result 
of selfish desires for comfort. It is not held together 
merely by economic necessity imposed upon separate 
individuals; not an aggregate of isolable constituents 
on which an external order is imposed. Even if it is 
organic, rather than spiritual, in so far as it is a whole 
constituted of parts, at any rate the interlocking of its 
parts is much more complex and vital than the relat- 
tionship obtaining between the members of the 'city of 
pigs'. 1 The 'real' social bond, here, turns out to be 

1 Republic II, 3720, 

spiritual being of man in his entirety. It goes beyond 
economics; it is more than an adjustment of demand 
and supply. And society reveals itself as the co-ordin- 
ation of subordinate totalities, each unique and very 
complicated, into a concrete whole. 

Yet without the earlier, imperfect analyses, Plato's 
final conception could not have been formed, and it 
would have lacked solidity and clearness. Not only do 
the erroneous and imperfect suggestions throw into 
sharper relief the more adequate account, but this 
more adequate theory takes up, incorporates and trans- 
forms the elements of the preliminary and one-sided 

And even Plato's own theory leaves much to be 
desired and points beyond itself to a more satisfactory 
conclusion. He represents the soul on the analogy 
rathef of an organic than of a spiritual whole. The 
corresponding conception of the state as the soul writ 
large gives us the same impression. The three classes in 
the state (as he describes them) are (or seem to be) 
integrants, though indispensable and inseparable parts. 
His theory thus has features that are not relevant to a 
genuinely philosophical theory of the state, which 
must reveal the spiritual real the soul with implicates 
as moments, and the state likewise as the soul writ 
large. To distinguish three classes in the state is essen- 
tial to a philosophical account only if and because 
they are identified with certain functions and activi- 
ties, not with certain groups of persons. They are 
moments in the life the spiritual life of man; not, 
strictly speaking, three estates, but the functions to 
which these correspond. Expressed precisely, there- 
fore, the jpojnents which Plato distinguishes are the 


wise administration of the laws (rather than the 
'Guardians'), the courageous upholding of the laws 
(rather than the 'Auxiliaries'), and the conscientious 
producing of the necessities of life (rather than the 



Absolute, 66, 6/ff. 
Adam, C, 13, 17. 
Aristotle, 32, 44f, 67, 69, 79, 83, 

109, nsf. 
Austin, Prof. J., 5, 82 113, 65 n2. 

Bacon, F., 55, 68. 
Baillet, 16, 18, 65 nl. 
Beck, Dr. L. J., 7. 
Belief, 30. 

Chanut, 15. 
Clerselier, 15, 16. 

Diophantus, 80. 
Descartes, Rene passim. 
Discours de la Methods, 13, 
14, 51, 75. 78, 80, 82, 83, 
84, 88, 93. 
La Dioptrique, 60. 
La Geometric, 94. 
La Recherche de la Verite, 13, 


Le Monde, 13, 15. 
L'Homme, 15. 
Meditationes, 13, 21. 
Principia Philosophiae, 13, 21, 

Regulae ad Directionem In- 

genii, passim. 

Deductio, 25, 28, 31, 35, 37-48, 
49, 54, 57, 59, 63, 77. 

Enumeratio see Inductio. 

Gilson, E., 17. 
Glazemaker, 17. 
Gouhier, H., 7. 

Hamelin, O., 80 ni. 
Heath, Sir T. L., 80 ni. 
Hegel, G W. F. f 115. 

lllatio, 25, 27, 37ff. 
Imagination (imaginatio), 2iff, 

28ff, in vera mathesis, 84ff, 

94, 96ff, see also phantasia. 
Implicates, ii2ff. 
Imitatio, 59, 61, 88. 
Inductio sive enumeratio, 38 ni. 

49-61, 57- 
Integrant, no-i. 

Intellect, 21, 28ff, 63, 64, 77, 94. 
Intuitus, 25, 27, 28-36, 37, 49, 

54. 63, 64, 77. 

Kant, I., 82 ns, 115, 118-9. 
Knowing, see vis cognoscens. 

Legrand, Abb Jean Baptiste, 15, 
1 6. 

Leibniz, G., 16, 17, 18, H5n, 

Leroy, 7. 

Liard, L., 76, ni, 93 n2, 94 n2. 
Locke, John, 107. 

Malebranche, N., 16. 
Marmion, 15, 16. 
Mathematics, 25, 73!?, 80-99, see 

also vera mathesis. 
Memory, 21, 50-1, 55, 96. 
Moment, 109, 120. 

Nature (natura) 68, naturae 
simplices, 3 iff, 70. 

Pappus, 79-80. 
Phantasia, 23, 97f. 
Plato, 11511, 120-2. 
Poisson, N., 1 6. 
Port Royal Logic, 16. 

Raey, Jean de, 14, 16. 
Relative (respective!) 676. 

Schuller, Dr., 16. 
Schopenhauer, A., 77. 
Sensation, sense, 2 iff, 28-9. 

sense-perception, 96-7. 

sensus communis, 22, 23, 29, 


Simple natures, see Nature. 
Spinoza, B. de, 45!, 68 ni. 

Tschirnhaus, E. W., 16. 

Vera Mathesis (mathesis univer- 

salis) 14, 25, 7sff, 80-99. 
Vieta, F., 80 ni. 
Vis cognoscens Ch. I, 63, g6L 

Will, 30. 


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